A few example designs and data sets for this module are available in the R package apts.doe, which can be installed from GitHub

library(devtools)
install_github("statsdavew/apts.doe", quiet = T)
library(apts.doe)

References will be provided throughout but some good general purpose texts are

• Atkinson, Donev and Tobias (2007). Optimum Experimental Design, with SAS. OUP
• Wu and Hamada (2009). Experiments: Planning, Analysis, and Parameter Design Optimization (2nd ed.). Wiley.
• Morris (2011). Design of Experiments: An Introduction based on Linear Models. Chapman and Hall/CRC Press.
• Santner, Williams and Notz (2019). The Design and Analysis of Computer Experiments (2nd ed.). Springer.

These notes and other resources can be found at https://statsdavew.github.io/apts.doe/

• Observational studies
• Sample surveys
• Designed experiments

Definition: An experiment is a procedure whereby controllable factors, or features, of a system or process are deliberately varied in order to understand the impact of these changes on one or more measurable responses.

See Luca and Bazerman (2020) for further history, annecdotes and examples, especially from psychology and technology.

Why do we experiment?

• key to the scientific method
  (hypothesis – experiment – observe – infer – conclude)

• potential to establish causality …

• … and to understand/improve complex systems depending on many factors

• comparison of treatments, factor screening, prediction, optimisation, …

Design of experiments: a statistical approach to the arrangement of the operational details of the experiment (eg sample size, specific experimental conditions investigated, …) so that the quality of the answers to be derived from the data is as high as possible.

1. Multi-factor experiment in pharmaceutical development.

Key to developing new medicines is the identification of optimal and robust process conditions (e.g. settings of temperature, pressure etc.) at which the active pharmaceutical ingredient should be synthesized.

[Somewhat confusinging, the FDA refer to this as identification of a “design space.”]

An important step in is this methodology is a robustness experiment to assess the sensitivity of identified conditions to changes in all (or at least very many) controllable factors.

While developing a new melanoma drug, GlaxoSmithKline performed an experiment to investigate sensitivity to 20 factors. Their experimental budget allowed only 10 individual experiments (runs) to be performed.

2. Computer experiments to optimise ride performance in luxury cars

Suspension settings can be used to improve the ride performance in cars. Optimising settings across many different car models would take many hundreds of hours of testing, so computer simulations are used.

Jaguar-Land Rover wanted to find suspension settings robust across different car models using a computer experiment (KTN workshop).

3. Optimal design to calibrate a physical model.

Physical (mechanistic, mathematical, …) models are used in many scientific fields. Typically, they are derived from fundamental understanding of the physics, chemistry, biology …

Most commonly, these models are solutions to differential equations. The models usually contain unknown parameters that should be estimated from experimental data.

Biologists at Southampton were studying the transfer of amino acids between mother and baby through the placenta. They could control the times at which observations were taken and the initial concentrations of amino acids (see Overstall, Woods, and Parker 2019).

Consider an experiment to compare two treatments (eg drugs, diets, fertilisers, \(\ldots\)).

We have \(n\) subjects (eg people, mice, plots of land, \(\ldots\)), each of which can be assigned to one of the two treatments.

A response (eg protein measurement, weight, yield, \(\ldots\)) is then measured from each subject.

Question: How should the two treatments be assigned to the subjects to gain the most precise inference about the difference in expected response from the two treatments.

Assume a linear model for the response \[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i\,,\qquad i=1,\ldots,n\,, \] with \(\varepsilon_i\sim N(0, \sigma^2)\) independently, \(\beta_0,\beta_1\) unknown parameters and \[ x_i = \left\{ \begin{array}{cc} -1 & \mbox{if treatment 1 is applied to subject $i$}\,, \\ +1 & \mbox{if treatment 2 is applied to subject $i$} \end{array} \right. \] The difference in expected response between treatment 1 and 2 is \[ E(y_i\,|\, x_i = +1) - E(y_i\,|\, x_i = -1) = \beta_0 + \beta_1 - \beta_0 + \beta_1 = 2\beta_1 \] So we need the most precise possible estimator of \(\beta_1\)

Both \(\beta_0\) and \(\beta_1\) can be estimated using least squares (or equivalently maximum likelihood).

Writing \[ \boldsymbol{y}= X\boldsymbol{\beta}+ \boldsymbol{\varepsilon}\,, \] we obtain estimators \[ \hat{\boldsymbol{\beta}} = \left(X^\mathrm{T}X\right)^{-1}X^\mathrm{T}\boldsymbol{y} \] with \[ \mbox{Var}(\hat{\boldsymbol{\beta}}) = \left(X^\mathrm{T}X\right)^{-1}\sigma^2 \] In this simple example, we are interesting in estimating \(\beta_1\), and we have \[ \begin{split} \mbox{Var}(\hat{\beta_1}) & = \frac{n\sigma^2}{n\sum x_i^2 - \left(\sum x_i\right)^2}\\ & = \frac{n\sigma^2}{n^2 - \left(\sum x_i\right)^2} \end{split} \]

Hence, we need to pick \(x_1,\ldots,x_n\) to minimise \(\left(\sum x_i\right)^2 = (n_1 - n_2)^2\)

• denote as \(n_1\) the number of subjects assigned to treatment 1, and \(n_2\) the number assigned to treatment 2, with \(n_1+n_2 = n\)
• it is obvious that \(\sum x_i = 0\) if and only if \(n_1 = n_2\)

Assuming \(n\) is even, the “optimal design” has \(n_1 = n_2 = n/2\)

For \(n\) odd, let \(n_1 = \frac{n+1}{2}\) and \(n_2 = \frac{n-1}{2}\)

We can assess a designs, labelled \(\xi\), via its efficiency relative to the optimal design \(\xi^\star\): \[ \mbox{Eff($\xi$)} = \frac{\mbox{Var}(\hat{\beta_1}\,|\,\xi^\star)}{\mbox{Var}(\hat{\beta_1}\,|\,\xi)} \]

n <- 50
eff <- function(n1) 1 - ((2 * n1 - n) / n)^2
curve(eff, from = 0, to = n, ylab = "Eff", xlab = expression(n[1]))

• Treatment – entities of scientific interest to be studied in the experiment
  eg varieties of crop, doses of a drug, combinations of temperature and pressure

• Unit – smallest subdivision of the experimental material such that two units may receive different treatments
  eg plots of land, subjects in a clinical trial, samples of reagent

• Run – application of a treatment to a unit

An initial step in fabricating integrated circuits is the growth of an epitaxial layer on polished silicon wafers via chemical deposition (see Wu and Hamada 2009, p155).

\[ y_{ij} = \mu + \tau_i + \varepsilon_{ij}\,,\qquad i=1,\ldots,t;\,j=1,\ldots,n_i\,, \] where

• \(y_{ij}\) : measured response from the \(j\)th unit to which treatment \(i\) has been applied

• \(\mu\) : overall mean response (often labelled \(\beta_0\))

• \(\tau_i\) : treatment effect (\(\tau_i\) is the expected difference in response from the overall mean after application of the \(i\)th treatment)

• \(\varepsilon_{ij}\) : random deviation from the expected response [typically \(\sim N(0,\sigma^2)\)]

The aims of the experiment are achieved by estimating comparisons between the treatment effects, \(\tau_k - \tau_l\).

Experimental precision and accuracy are largely obtained through control and comparison.

Three key model assumptions are:

• additivity (response = treatment effect + unit effect)
• constancy of treatment effects (treatment effect does not depend on the unit to which it is applied)
• no interference between units (the effect of a treatment applied to unit \(j\) does not depend on the treatment applied to any other unit)

See Dasgupta, Pillai, and Rubin (2015) for discussion of these assumptions for factorial experiments

Stratification (blocking)

• account for systematic differences between batches of experimental units by arranging them in homogeneous sets (blocks)
  • if the same treatment was applied to all units, within-block variation in the response would be much less than between-block
  • compare treatments within the same block and hence eliminate block effects

Replication

• the application of each treatment to multiple experimental units
  • provides an estimate of experimental error against which to judge treatment differences
  • reduces the variance of the estimators of treatment differences

Randomisation

• we randomise features such as the allocation of units to treatments, the order in which treatments are applied, …
  • protects against lurking (uncontrolled) variables (model-robust) and subjectively in the allocation of treatments to units

Randomisation is perhaps the key principle in the design of experiments

• it protects against model misspecification (bias), and hence allows causality to be established
  • a clear difference between treatments can only be an accident of the randomisation or a consequence of the treatments
• unbiased estimation of \(\tau\) and \(\sigma^2\), even if the errors are not normally distributed
• exact tests for differences between treatment effects are available (Basu 1980)

Without randomisation, unobserved confounders (\(U\)) can induce a dependency between
the response (\(Y\)) and treatment (\(T\)) cf Cox and Reid (2000), p.35

With randomisation, unobserved confounders (\(U\)) are independent of the treatment (\(T\)). Marginalisation over \(U\) does not induce an edge between \(T\) and \(Y\) cf Cox and Reid (2000), p.35

Fabrication of integrated circuits (Wu and Hamada 2009, p155)

Treatment

• combination of settings of the factors
  • A : rotation method (\(x_1\))
  • B : nozzle position (\(x_2\))
  • C : deposition temperature (\(x_3\))
  • D : deposition time (\(x_4\))

Assume each factor has two-levels, coded -1 and +1

Each factor has two levels \(x_k = \pm 1,\, k=1,\ldots,4\)

A treatment is then defined as a combination of four values of \(-1, +1\)

• eg \(x_1 = -1, x_2 = -1, x_3 = +1, x_4 = -1\)
• specifies a setting of the process

Assume each treatment effect is determined by a regression model in the four factors, eg \[ \tau(\boldsymbol{x}) = \sum_{i=1}^4\beta_ix_i + \sum_{j=1}^4\sum_{i>j}^4\beta_{ij}x_ix_j \]

with(cirfab, cirfab[order(x1, x2, x3, x4), ])

##    x1 x2 x3 x4     ybar
## 2  -1 -1 -1 -1 13.58983
## 1  -1 -1 -1  1 14.59000
## 4  -1 -1  1 -1 14.04983
## 3  -1 -1  1  1 14.24000
## 6  -1  1 -1 -1 13.94000
## 5  -1  1 -1  1 14.65000
## 8  -1  1  1 -1 14.14017
## 7  -1  1  1  1 14.40000
## 10  1 -1 -1 -1 13.72000
## 9   1 -1 -1  1 14.67000
## 12  1 -1  1 -1 13.90000
## 11  1 -1  1  1 13.84017
## 14  1  1 -1 -1 13.87983
## 13  1  1 -1  1 14.56000
## 16  1  1  1 -1 14.11017
## 15  1  1  1  1 14.30000

• treatments in standard order

• \(\bar{y}\) - average response from the six wafers

\[ \boldsymbol{Y} = X\boldsymbol{\beta} + \boldsymbol{\varepsilon}\,,\qquad \boldsymbol{\varepsilon}\sim N(\boldsymbol{0}, \sigma^2I)\,,\qquad \hat{\boldsymbol{\beta}} = \left(X^\mathrm{T}X\right)^{-1}X^\mathrm{T}\boldsymbol{Y} \]

• model matrix \(X\) has columns corresponding to intercept, linear and cross-product terms

• information matrix \(X^\mathrm{T}X = nI\)

• regression coefficients are estimated by independent contrasts in the data

cirfab.lm <- lm(ybar ~ (.) ^ 2, data = cirfab)
coef(cirfab.lm)

##  (Intercept)           x1           x2           x3           x4        x1:x2 
## 14.161250000 -0.038729167  0.086270833 -0.038708333  0.245020833  0.003708333 
##        x1:x3        x1:x4        x2:x3        x2:x4        x3:x4 
## -0.046229167 -0.025000000  0.028770833 -0.015041667 -0.172520833

Main effect of \(x_k\): \[ [\text{Avg. response when $x_k = 1$}]\, -\, [\text{Avg. response when $x_k = -1$}] \]

Interaction between \(x_j\) and \(x_k\): \[ [\text{Avg. response when $x_jx_k= 1$}]\, -\, [\text{Avg. response when $x_jx_k = -1$}] \]

Higher-order interactions defined similarly

Assuming -1,+1 coding, there is a straightforward relationship between factorial effects and regression coefficients

• main effect of \(x_k\) is equal to \(2\beta_k\)
• interaction between \(x_j\) and \(x_k\) is equal to \(2\beta_{jk}\)

Using the effects package:

library(effects)
plot(Effect("x1", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x2", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x3", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x4", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)

plot(Effect(c("x3", "x4"), cirfab.lm, xlevels = 2), main = "", rug = F, ylim = c(13.5, 15), 
     x.var = "x4")

\(X^\mathrm{T}X = nI \Rightarrow \hat{\boldsymbol{\beta}}\) are independently normally distributed with equal variance

Hence, we can treat the identification of important effects (ie large \(\beta\)) as an outlier identification problem

• plot (absolute) ordered factorial effects against (absolute) quantiles from a standard normal
• outlying effects are identified as important

Cuthbert (1959)

Using the FrF2 package

library(FrF2)
par(pty = "s", mar = c(8, 4, 1, 2))
DanielPlot(cirfab.lm, main = "", datax = F, half = T)

An unreplicated factorial design provides no model-independent estimate of \(\sigma^2\) (Gilmour and Trinca 2012)

• any unsaturated model does provide an estimate, but it may be biased by ignored (significant) model terms
• this is one reason why graphical (or associated) analysis methods are popular

Replication also increases the power of the design

• common to replicate a centre point
• allows a portmanteau test of curvature

Refit a linear model to the fabricated circuits experiment excluding interactions.

cirfab2.lm <- lm(ybar ~ (.), data = cirfab)
coef(cirfab.lm)

##  (Intercept)           x1           x2           x3           x4        x1:x2 
## 14.161250000 -0.038729167  0.086270833 -0.038708333  0.245020833  0.003708333 
##        x1:x3        x1:x4        x2:x3        x2:x4        x3:x4 
## -0.046229167 -0.025000000  0.028770833 -0.015041667 -0.172520833

coef(cirfab2.lm)

## (Intercept)          x1          x2          x3          x4 
## 14.16125000 -0.03872917  0.08627083 -0.03870833  0.24502083

cbind(sigma1 = sigma(cirfab.lm), df1 = df.residual(cirfab.lm),  
      sigma2 = sigma(cirfab2.lm), df2 = df.residual(cirfab2.lm))

##        sigma1 df1    sigma2 df2
## [1,] 0.137152   5 0.2396108  11

Effect sparsity

• the number of important effects in a factorial experiment is small relative to the total number of effects investigated (cf Box and Meyer 1986)

Effect hierarchy

• lower-order effects are more likely to be important than higher-order effects
• effects of the same order are equally likely to be important

Effect heredity

• interactions where at least one parent main effect is important are more likely to be important themselves

Wu and Hamada (2009), pp.172–172

Factorial designs can require a large number of runs for only a moderate number of factors (\(2^5 = 32\))

Resource constraints (eg cost) may mean not all \(2^m\) combinations can be run

Lots of degrees of freedom are devoted to estimating higher-order interactions

• eg in a \(2^5\) experiment, 16 degrees of freedom are used to estimate three-factor and higher-order interactions
• principles of effect hierarchy and sparsity suggest this may be wasteful

Need to trade-off what you need to estimate against the number of runs you can afford

Production of bacteriocin, a targetted antibacterial used as a natural food preservative (Morris 2011, p231)

Assume each factor has two-levels, coded -1 and +1

Find an \(n=8\) run design using FrF2

bact.design <- FrF2(8, 5, factor.names = paste0("x", 1:5), 
     generators = list(c(1, 3), c(2, 3)), randomize = F, alias.info = 3)
bact.design

##   x1 x2 x3 x4 x5
## 1 -1 -1 -1  1  1
## 2  1 -1 -1 -1  1
## 3 -1  1 -1  1 -1
## 4  1  1 -1 -1 -1
## 5 -1 -1  1 -1 -1
## 6  1 -1  1  1 -1
## 7 -1  1  1 -1  1
## 8  1  1  1  1  1
## class=design, type= FrF2.generators

• \(8\) = \(32/4\) = \(2^5/2^2\) = \(2^{5-2}\)
• we need a principled way of choosing one-quarter of the runs from the factorial design that leads to clarity in the analysis

Assuming the number of runs is a power of two, \(n = 2^{k-q}\), we can construct \(2^{k-q} -1\) orthogonal vectors (with inner product zero), spanned by \(k-q = \log_2(n)\) vectors

• construct the full factorial design for \(k-q\) factors
• assign the remaining \(q\) factors to interaction columns

model.matrix(~ (x1 + x2 + x3) ^ 3, bact.design[, 1:3])[, -1]

##   x11 x21 x31 x11:x21 x11:x31 x21:x31 x11:x21:x31
## 1  -1  -1  -1       1       1       1          -1
## 2   1  -1  -1      -1      -1       1           1
## 3  -1   1  -1      -1       1      -1           1
## 4   1   1  -1       1      -1      -1          -1
## 5  -1  -1   1       1      -1      -1           1
## 6   1  -1   1      -1       1      -1          -1
## 7  -1   1   1      -1      -1       1          -1
## 8   1   1   1       1       1       1           1

The design has been deliberately chosen so that

• \(x_4 = x_1x_3\)
• \(x_5 = x_2x_3\)

[\(x_1x_2\) is shorthand for the Hadamard (Schur or entry wise) product of two vectors, \(x_1\circ x_2\)]

What other consequences are there?

• \(x_4x_5 = x_1x_3x_2x_3 = x_1x_2x_3^2\)
• the product of any column with itself is the constant column (the identity)
• hence, \(x_4x_5 = x_1x_2\)

Now we can obtain the defining relation \(\ldots\)

• \(I = x_1x_3x_4 = x_2x_3x_5 = x_1x_2x_4x_5\)

\(\ldots\) and the complete aliasing scheme

• \(x_1 = x_3x_4 = x_1x_2x_3x_5 = x_2x_4x_5\)
• \(x_2 = x_1x_2x_3x_4 = x_3x_5 = x_1x_4x_5\)
• \(x_3 = x_1x_4 = x_2x_5 = x_1x_2x_3x_4x_5\)
• \(x_4 = x_1x_3 = x_2x_3x_4x_5 = x_1x_2x_5\)
• \(x_5 = x_1x_3x_4x_5 = x_2x_3 = x_1x_2x_4\)
• \(x_1x_2 = x_2x_3x_4 = x_1x_3x_5 = x_4x_5\)
• \(x_1x_5 = x_3x_4x_5 = x_1x_2x_3 = x_2x_4\)

FrF2 will summarise the aliasing amongst main effects and two- and three-factor interactions.

design.info(bact.design)$aliased 

## $legend
## [1] "A=x1" "B=x2" "C=x3" "D=x4" "E=x5"
## 
## $main
## [1] "A=CD=BDE" "B=CE=ADE" "C=AD=BE"  "D=AC=ABE" "E=BC=ABD"
## 
## $fi2
## [1] "AB=DE=ACE=BCD" "AE=BD=ABC=CDE"
## 
## $fi3
## [1] "ACD=BCE"

What is the consequence of this aliasing?

If more than one effect in each alias string is non-zero, the least squares estimators will be biased

• assumed model \(\boldsymbol{Y}= X_1\boldsymbol{\beta}_1 + \boldsymbol{\varepsilon}\)
• true model \(\boldsymbol{Y}= X_1\boldsymbol{\beta}_1 + X_2\boldsymbol{\beta}_2 + \boldsymbol{\varepsilon}\)

\[ \begin{split} E\left(\hat{\boldsymbol{\beta}}_1\right) & = \left(X_1^\mathrm{T}X_1\right)^{-1}X^\mathrm{T}_1E(\boldsymbol{Y}) \\ & = \left(X^\mathrm{T}_1X_1\right)^{-1}X_1^\mathrm{T}\left(X_1\boldsymbol{\beta}_1 + X_2\boldsymbol{\beta}_2\right) \\ & = \beta_1 + \left(X_1^\mathrm{T}X_1\right)^{-1}X_1^\mathrm{T}X_2\boldsymbol{\beta}_2 \\ & = \boldsymbol{\beta}_1 + A\boldsymbol{\beta}_2\\ \end{split} \]

\(A\) is the alias matrix

• if the columns of \(X_1\) and \(X_2\) are not orthogonal, \(\hat{\boldsymbol{\beta}}_1\) is biased

For the \(2^{5-2}\) example considering “main effects” and “two-factor interactions”:

• \(X_1\) is an \(8\times 8\) matrix with columns for the intercept, five linear and two product terms
• \(X_2\) is an \(8\times 8\) matrix with columns for the 8 remaining product terms

The transpose of the alias matrix is provided by the alias function.

ff.alias <- alias(y ~ (.)^2, data = data.frame(bact.design, y = vector(length = 8)))
t(ff.alias$Complete)[, order(rownames(ff.alias$Complete))]

##             x11:x31 x11:x41 x21:x31 x21:x41 x21:x51 x31:x41 x31:x51 x41:x51
## (Intercept) 0       0       0       0       0       0       0       0      
## x11         0       0       0       0       0       1       0       0      
## x21         0       0       0       0       0       0       1       0      
## x31         0       1       0       0       1       0       0       0      
## x41         1       0       0       0       0       0       0       0      
## x51         0       0       1       0       0       0       0       0      
## x11:x21     0       0       0       0       0       0       0       1      
## x11:x51     0       0       0       1       0       0       0       0

For a regular design, the matrix \(A\) will only have entries 0, \(\pm 1\) (no aliasing or complete aliasing)

These linear dependencies can be seen if we attempt to fit a linear model

bact.lm <- lm(yB ~ (x1 + x2 + x3 + x4 + x5)^2, data = bact)
summary(bact.lm)

## 
## Call:
## lm.default(formula = yB ~ (x1 + x2 + x3 + x4 + x5)^2, data = bact)
## 
## Residuals:
## ALL 8 residuals are 0: no residual degrees of freedom!
## 
## Coefficients: (8 not defined because of singularities)
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  4.42625        NaN     NaN      NaN
## x1           0.26625        NaN     NaN      NaN
## x2           0.24625        NaN     NaN      NaN
## x3          -0.22875        NaN     NaN      NaN
## x4          -1.11875        NaN     NaN      NaN
## x5          -0.66375        NaN     NaN      NaN
## x1:x2       -0.05375        NaN     NaN      NaN
## x1:x3             NA         NA      NA       NA
## x1:x4             NA         NA      NA       NA
## x1:x5       -0.13375        NaN     NaN      NaN
## x2:x3             NA         NA      NA       NA
## x2:x4             NA         NA      NA       NA
## x2:x5             NA         NA      NA       NA
## x3:x4             NA         NA      NA       NA
## x3:x5             NA         NA      NA       NA
## x4:x5             NA         NA      NA       NA
## 
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:    NaN 
## F-statistic:   NaN on 7 and 0 DF,  p-value: NA

Typically, in a first experiment, fractional factorial designs are used in screening

• investigate which of many factors have a substantive effect on the response
• main effects and two-factor interactions
• centre points to check for curvature

At second and later stages, augment the design

• to resolve ambiguities due to the aliasing of factorial effects (“break the alias strings”)
• to allow estimation of curvature and prediction from a more complex model

Regular fractional factorial designs have the number of runs equal to a power of the number of levels

• eg \(2^{5-2}\), \(3^{3-1}\times 2\)
• this inflexibility in run sizes can be a problem in practical experiments

Non-regular designs can have any number of runs (usually with \(n>p\), the number of parameters to be estimated)

Often the clarity provided by a regular design is lost

• no defining relation or straightforward aliasing scheme
• partial aliasing and fractional entries in \(A\)

One approach to finding non-regular designs is via a design optimality criterion

Notation: let \(\xi = [\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n]\) denote a design (choice of treatments and their replications)

Assuming the model \(\boldsymbol{Y}= X\boldsymbol{\beta}+ \boldsymbol{\varepsilon}\), with \(\boldsymbol{\varepsilon}\sim N(0, \sigma^2I_n)\), a \(D\)-optimal design maximises \[ \phi(\xi) = \mathrm{det}\left(X^\mathrm{T}X\right) \]

That is, a \(D\)-optimal design maximises the determinant of the (expected) Fisher information matrix

• equivalent to minimising the volume of the joint confidence ellipsoid for \(\boldsymbol{\beta}\)

Also useful to define a Bayesian version, with \(R\) a prior precision matrix \[ \phi_B(\xi) = \mathrm{det}\left(X^\mathrm{T}X + R\right) \] (See later)

\(D\)-optimal designs are model dependent

• if the model (ie the columns of \(X\)) changes, the optimal design may change
• model-robust design is an active area of research

\(D\)-optimality promotes orthogonality in the \(X\) matrix

• if there are sufficient runs, the \(D\)-optimal design will usually be orthogonal
• for particular models and choices of \(n\), regular fractional factorial designs are \(D\)-optimal

There are many other optimality criteria, tailored to other experimental goals

• prediction, model discrimination, space-filling, …

\(k=11\) factors in \(n=12\) runs, first-order (main effects) model (Plackett and Burman 1946)

A particular \(D\)-optimal design is the following orthogonal array

Using the pb function in the FrF2 package:

pb.design <- pb(12, factor.names = paste0("x", 1:11))
pb.design

##    x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
## 1   1  1  1 -1 -1 -1  1 -1  1   1  -1
## 2  -1  1 -1  1  1 -1  1  1  1  -1  -1
## 3   1 -1 -1 -1  1 -1  1  1 -1   1   1
## 4   1  1 -1  1  1  1 -1 -1 -1   1  -1
## 5  -1  1  1 -1  1  1  1 -1 -1  -1   1
## 6  -1 -1  1 -1  1  1 -1  1  1   1  -1
## 7   1 -1  1  1 -1  1  1  1 -1  -1  -1
## 8   1  1 -1 -1 -1  1 -1  1  1  -1   1
## 9  -1 -1 -1  1 -1  1  1 -1  1   1   1
## 10  1 -1  1  1  1 -1 -1 -1  1  -1   1
## 11 -1 -1 -1 -1 -1 -1 -1 -1 -1  -1  -1
## 12 -1  1  1  1 -1 -1 -1  1 -1   1   1
## class=design, type= pb

This 12-run PB design is probably the most studied non-regular design

• orthogonal columns
• complex aliasing between main effects and two-factor interactions

pb.alias <- alias(y ~ (.)^2, data = data.frame(pb.design, y = vector(length = 12)))
t(pb.alias$Complete)[, 1:8]

##             x11:x21 x11:x31 x11:x41 x11:x51 x11:x61 x11:x71 x11:x81 x11:x91
## (Intercept)    0       0       0       0       0       0       0       0   
## x11            0       0       0       0       0       0       0       0   
## x21            0    -1/3    -1/3    -1/3     1/3    -1/3    -1/3     1/3   
## x31         -1/3       0     1/3    -1/3    -1/3     1/3    -1/3     1/3   
## x41         -1/3     1/3       0     1/3     1/3    -1/3    -1/3    -1/3   
## x51         -1/3    -1/3     1/3       0    -1/3    -1/3    -1/3    -1/3   
## x61          1/3    -1/3     1/3    -1/3       0    -1/3     1/3    -1/3   
## x71         -1/3     1/3    -1/3    -1/3    -1/3       0     1/3    -1/3   
## x81         -1/3    -1/3    -1/3    -1/3     1/3     1/3       0    -1/3   
## x91          1/3     1/3    -1/3    -1/3    -1/3    -1/3    -1/3       0   
## x101         1/3    -1/3    -1/3     1/3    -1/3     1/3    -1/3    -1/3   
## x111        -1/3    -1/3    -1/3     1/3    -1/3    -1/3     1/3     1/3

Screening designs with fewer runs than factors (see Woods and Lewis 2017)

• can’t use ordinary least squares/maximum likelihood as \(X\) does not have full column rank
• Bayesian \(D\)-optimality with \(R = [0\,|\, \tau I_m]\)

Supersaturated experiment used by GlaxoSmithKline in the development of a new oncology drug

• \(k=16\) factors: e.g. temperature, solvent amount, reaction time
• \(n=10\) runs
• Bayesian \(D\)-optimal design with \(\tau = 0.2\)

ssd

##    x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16
## 1   1  1 -1  1  1 -1 -1 -1 -1  -1  -1   1   1   1   1   1
## 2   1  1  1 -1 -1 -1 -1 -1  1   1  -1  -1   1  -1  -1   1
## 3  -1 -1  1 -1  1 -1  1 -1 -1   1  -1  -1  -1   1   1  -1
## 4  -1  1  1  1  1 -1 -1  1 -1  -1   1  -1  -1  -1  -1  -1
## 5  -1 -1 -1 -1 -1  1 -1  1  1  -1  -1  -1  -1  -1   1   1
## 6   1  1  1 -1  1  1 -1  1  1   1   1   1   1   1   1  -1
## 7  -1 -1  1  1 -1 -1  1 -1  1  -1   1   1   1  -1   1  -1
## 8   1 -1 -1  1  1  1  1  1 -1   1  -1   1   1  -1  -1  -1
## 9  -1  1 -1  1 -1  1  1 -1  1   1   1  -1  -1   1  -1   1
## 10  1 -1  1 -1 -1 -1  1  1 -1  -1   1   1  -1   1  -1   1

Partial aliasing between main effects

Heatmap of column correlations:

library(fields)
par(mar=c(8,2,0,0))
image.plot(1:16,1:16, cor(ssd), zlim = c(-1, 1), xlab = "Factors", 
           ylab = "", asp = 1, axes = F)
axis(1, at = seq(2, 16, by = 2), line = .5)
axis(2, at = seq(2, 16, by = 2), line = -5)

Analysis via regularised (shrinkage) methods (eg lasso, Dantzig selector; see APTS High Dimensional Statistics)

• small coefficients shrunk to zero

Many physical and social processes can be approximated by computer codes which encapsulate mathematical models

• eg partial differential equations solved using finite element methods
• eg reaction kinetics modelling in computational biology, in-silico chemistry

- computer code: numerical implementation of the mathematical model

Key feature: the model does not have a closed-form; it can only be evaluated numerically, and this is typically (relatively) expensive

We will focus on deterministic computer models

Assumption: \(g(\boldsymbol{x})\) can only be evaluated numerically; i.e. \(g(\boldsymbol{x})\) can be computed for a given \(\boldsymbol{x}\) but the general form is unknown

How do we learn about the function \(g(\boldsymbol{x})\)?

In an analogy to a physical system, we experiment on \(g(\boldsymbol{x})\), i.e.

• choose a design \(\xi = (\boldsymbol{x}_1,\ldots, \boldsymbol{x}_n)\)
• evaluate \(g(\boldsymbol{x}_i)\) (run the computer code)

Use the “data” \(\left\{\boldsymbol{x}_i, g(\boldsymbol{x}_i)\right\}\) to build statistical models linking \(\boldsymbol{x}\) and \(g(\boldsymbol{x})\)

• called emulators; typically use a Gaussian process

See Santner, Williams, and Notz (2019)

Climate modelling involves the solution of many intractable equations, leading to mathematical models evaluated via computationally expensive computer codes

• lots of applications of computer experiments

We will illustrate methods on a very simple example: a time-stepping advective/diffusive surface layer meridional EBM (energy balance model)

• 2D earth with no land
• each surface object has a percentage of ice cover
• different albedo (fraction of solar energy reflected) for ice vs non-ice surfaces
• ocean circulation is explicitly modelled (cf Atlantic gulf stream)
• two variables: \(x_1\) - solar constant; \(x_2\) - non-ice albedo
• output is mean temperature

See https://wiki.aston.ac.uk/foswiki/bin/view/MUCM/SurfebmModel (temporarily unavailable - hopefully back soon!)

## design and data are in 'ebm'
library(akima)
fld <- interp(x = ebm$x1, y = ebm$x2, z = ebm$y, extrap = TRUE, linear = FALSE)
filled.contour(x = fld$x, y = fld$y, z = fld$z, ylim = c(-1, 1), xlim = c(-1, 1), asp = 1, 
               plot.title = title(xlab = "solar constant", ylab = "non-ice albedo"), 
               plot.axes = {axis(1, seq(-1, 1, l = 5))
                 axis(2, seq(-1, 1, l = 5))})

As we will see later, emulators are usually constructed using nonparametric statistical models

This choice leads naturally to using space-filling designs

• such designs do not rely on the functional form of the relationship between the code inputs and the response
• good coverage is important for prediction (we will predict “better” near points we have already run the computer model)

Common designs are chosen to optimise some space-filling metric, or formed from (stratified) random sampling

Space-filling designs do not have replication, so ideal for deterministic computer models

Many designs proposed for computer experiments are related to ideas underpinning quadrature, and the approximation of an expectation.

Let \(\bar{g} = \frac{1}{n}\sum_{i=1}^n g(\boldsymbol{x}_i)\), the sample mean of \(g(\cdot)\) for \(\xi\). Then

\[ |E_\boldsymbol{x}[g(\boldsymbol{x})] - \bar{g}| \le \mbox{constant}\times D(\xi) \] where \(D(\xi)\) is the star discrepancy of the design

• \(D(\xi)\) is a measure of the uniformity of the design points

This relationship leads to the criterion of design selection via minimising discrepancy

• \(D(\xi)\) is difficult to compute for moderate to high numbers of dimensions
• therefore, it is more common to minimise the related centred \(L_2\)-discrepancy

Fang, Li, and Sudjianto (2006), Ch.3

Two sensible criteria for the selection of a space-filling design are

• make sure no two points in the design are too close together
• make sure no point in the design region is too far from a design point

(Johnson, Moore, and Ylvisaker 1990)

The Euclidean distance between points \(\boldsymbol{x}\) and \(\boldsymbol{x}^\prime\) is given by

\[ \delta(\boldsymbol{x}, \boldsymbol{x}^\prime) = \sqrt{\sum_{i=1}^k \left(x_{j} - x^\prime_j\right)^2} \]

Using Euclidean distance, we can define

• maximin (Mm) criterion: maximise

\[ \min_{\boldsymbol{x}_i, \boldsymbol{x}_j\in\xi}\delta(\boldsymbol{x}_i, \boldsymbol{x}_j) \]

• minimax (mM) criterion: minimise

\[ \max_{\boldsymbol{x}}\delta(\boldsymbol{x}, \xi) \] where the distance between a point \(\boldsymbol{x}\) and a design \(\xi\) is defined as

\[ \delta(\boldsymbol{x}, \xi) = \min_{\boldsymbol{x}_j\in\xi}\delta(\boldsymbol{x}, \boldsymbol{x}_i) \]

Roughly speaking, an Mm design spreads out the design points, and an mM design covers the design region

Intuitively, covering the design region seems more desirable (eg for prediction), but optimising the mM objective function is computationally challenging. Hence, Mm designs are more commonly used

For high-dimensional problems, space-filling is difficult

• many points are required to adequate space-fill a high-dimensional space (curse of dimensionality)

Latin hypercube designs (LHDs) are randomly chosen sets of points with the restriction of uniform one-dimensional projections (McKay, Beckman, and Conover 1979)

• each variable has no overlapping points, and good coverage (compare with a factorial design, which has hidden replication)
• can be easily constructed using permutations of integers

An LHD only guarantees space-filling properties in each one-dimensional projection, not overall. So we normally combine the Latin hypercube principle with a space-filling criteria, eg to find a Mm LHD

LH <- function(n = 3, d = 2) {
    D <- NULL
    for(i in 1:d) D <- cbind(D, sample(1:n, n))
    D 
}
set.seed(4)
par(mar=c(5,6,2,4)+0.1, pty = "s")
plot((LH() - .5) / 3, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 1.5, cex.axis = 1, cex = 2)
abline(v = c(0, 1/3, 2/3, 1), lty = 2)
abline(h = c(0, 1/3, 2/3, 1), lty = 2)

The DiceDesign package has functions to generate various LHDs

library(DiceDesign)
lhs.d <- lhsDesign(9, 2)
plot(lhs.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "random", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

discrep.d <- discrepSA_LHS(lhs.d$design, criterion = "C2")
plot(discrep.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "discrepancy", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

maximin.d <- maximinSA_LHS(discrep.d$design)
plot(maximin.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "maximin", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

The design for the EBM example is a Mm LHD

par(mar=c(5,6,2,4)+0.1, pty = "s")
plot(ebm[, 2:3], xlim = c(-1, 1), ylim = c(-1, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, asp = 1)
abline(v = 2 * seq(0:20) / 20 - 1, lty = 2)
abline(h = 2 * seq(0:20) / 20 - 1, lty = 2)

The most common statistical model used to emulate computer models is the Gaussian process (GP)

• flexible, nonparametric regression model (few assumptions made about \(g(\boldsymbol{x})\))
• naturally allows for uncertainty quantification (eg prediction intervals)
• interpolates observed responses

An intuitive way to think about a GP is as a prior for the unknown function \(g(\boldsymbol{x})\) within a Bayesian framework

We say that

\[ g(\boldsymbol{x})\sim \text{GP}\left(\mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}, \sigma^2\kappa(\boldsymbol{x},\boldsymbol{x}^\prime;\,\boldsymbol{\theta})\right)\,, \] where \(\mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}\) is the mean, \(\kappa(\boldsymbol{x},\boldsymbol{x}^\prime;\,\boldsymbol{\phi})\) is the correlation function, \(\boldsymbol{\theta}\) is the vector of correlation parameters and \(\sigma^2\) is the constant variance, if:

• any vector \(\boldsymbol{g}= \left(g(\boldsymbol{x}_1), \dots , g(\boldsymbol{x}_n)\right)^{\mathrm{T}}\) satisfies \[\boldsymbol{g}\sim N\left(F\boldsymbol{\beta}, \sigma^2 K(\boldsymbol{\theta})\right)\,,\] with \(F\) a model matrix and \(K\) the \(m\times m\) covariance matrix defined by \(K(\boldsymbol{\theta})_{ij} = \kappa(\boldsymbol{x}_i,\boldsymbol{x}_j;\boldsymbol{\theta})\).

See Rasmussen and Williams (2006)

Typically, very simple mean functions are chosen for the GP, eg

• constant: \(\mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}= \beta_0\) (sometimes called ordinary kriging)
• linear: \(\mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}= \beta_0 + \sum_{j=1}^k\beta_jx_j\) (universal kriging)

The most commonly used correlation functions are separable and stationary

• squared exponential:

\[ \kappa(\boldsymbol{x}, \boldsymbol{x}^\prime;\,\boldsymbol{\theta})=\exp\left[-\sum_j\left(\frac{|x_{j} - x^\prime_{j} |}{\theta_j }\right)^2\right] \]

• Matérn \(\nu = 5/2\)

\[ \kappa(\boldsymbol{x}, \boldsymbol{x}^\prime; \,\boldsymbol{\theta}) = \prod_{j}\left(1 + \sqrt{5}\frac{|x_j - x_j^\prime|}{\theta_j} + \frac{5}{3}\left(\frac{|x_j - x_j^\prime|}{\theta_j}\right)^2\right)\exp\left(-\sqrt{5}\frac{|x_j - x_j^\prime|}{\theta_j}\right) \] The Matérn function can be defined for other values of \(\nu\); for \(\nu\rightarrow\infty\), the squared exponential function is obtained

Given model evaluations \(\boldsymbol{g}= \left[g(\boldsymbol{x}_1), \ldots, g(\boldsymbol{x}_n)\right]\), a posterior GP can be obtained:

\[ g(\boldsymbol{x})\,|\, \boldsymbol{g},\boldsymbol{\beta},\boldsymbol{\theta},\sigma^2 \sim \text{GP}\left(m(\boldsymbol{x}), s^2(\boldsymbol{x})\right) \]

• \(m(\boldsymbol{x}) = \mathbf{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}+ \boldsymbol{\kappa}_n^\mathrm{T}K^{-1}(\boldsymbol{g}- F\boldsymbol{\beta})\)
• \(s^2(\boldsymbol{x}) = \sigma^2\left(1 - \boldsymbol{\kappa}_n^\mathrm{T}K^{-1}\boldsymbol{\kappa}_n\right)\)

where \(\boldsymbol{\kappa}_n = [\kappa(\boldsymbol{x},\boldsymbol{x}_i\,;\,\boldsymbol{\theta})]_{i=1}^n\) is a vector of correlations between \(g(\boldsymbol{x})\) and \(g(\boldsymbol{x}_1),\ldots,g(\boldsymbol{x}_n)\)

The updating of the prior mean and variance depends on the “distance” between \(\boldsymbol{x}\) and the points in \(\xi\)

• the posterior mean will be adjusted more for points closer to the design
• predictions at these points will have smaller posterior variance

If \(\boldsymbol{x}= \boldsymbol{x}_i\) (so we are predicting at a design point), \(K^{-1}\boldsymbol{\kappa}_n = \boldsymbol{e}_i\), the \(i\)th unit vector

• \(m(\boldsymbol{x}_i) = \mathbf{f}(\boldsymbol{x}_i)^\mathrm{T}\boldsymbol{\beta}+ \boldsymbol{e}_i^\mathrm{T}(\boldsymbol{g}- F\boldsymbol{\beta}) = g(\boldsymbol{x}_i)\)
• \(s^2(\boldsymbol{x}_i) = \sigma^2\left(1 - \boldsymbol{\kappa}_n^\mathrm{T}\boldsymbol{e}_i\right) = \sigma^2\left(1 - \kappa(\boldsymbol{x}_i,\boldsymbol{x}_i\,;\,\boldsymbol{\theta})\right) = 0\)

The posterior GP interpolates - exactly what you want for a deterministic computer code

Inference unconditional on all the hyperparameters requires numerical approximation (eg Markov chain Monte Carlo)

• it is common to estimate the parameters, eg using maximum likelihood, to “plug-in” to the posterior predictive distribution

A simple example: \(g(x) = \sin(2\pi x)\) using the DiceKriging package

library(DiceDesign)
library(DiceKriging)
xi <- lhsDesign(6, 1)$design
y <- sin(2 * pi * xi)
gp <- km(design = xi, response = y, control = list(trace = F))
xs <- sort(c(seq(-.1, 1.1, length = 100), xi))
gpp <- predict(gp, newdata = xs, type = "SK")

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x")
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, sin(2 * pi * xs), lty = 1, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
legend(x = "topright", legend = c("posterior mean of g", "posterior quantiles for g", 
                                  expression(paste("observed data ", g(x[i]))), 
                                    "function g(.)"), lty = c(1, 2, NA, 1), 
       pch = c(NA, NA, 4, NA), lwd = c(2, 2, 2, 2), col = c("red", "black", "blue", "blue"))

Return to the EBM example

gpebm <- km(formula = ~., design = ebm[, 2:3], response = ebm[, 1], control = list(trace = F))
gpebm

## 
## Call:
## km(formula = ~., design = ebm[, 2:3], response = ebm[, 1], control = list(trace = F))
## 
## Trend  coeff.:
##                Estimate
##  (Intercept)    16.3262
##           x1     2.4078
##           x2   -28.9973
## 
## Covar. type  : matern5_2 
## Covar. coeff.:
##                Estimate
##    theta(x1)     2.8829
##    theta(x2)     0.2722
## 
## Variance estimate: 2.215045

xs1 <- sort(c(seq(-1, 1, length = 10), ebm[, 2]))
xs2 <- sort(c(seq(-1, 1, length = 10), ebm[, 3]))
xs <- expand.grid(x1 = xs1, x2 = xs2)
gppebm <- predict(gpebm, newdata = xs, type = "UK")
filled.contour(x = xs1, y = xs2, z = matrix(gppebm$mean, nrow = length(xs1)))
filled.contour(x = xs1, y = xs2, z = matrix(gppebm$sd, nrow = length(xs1)),
               plot.axis = {axis(1); axis(2); points(ebm[, 2:3])})

A common task is optimisation of \(g(\boldsymbol{x})\)

When \(g(\boldsymbol{x})\) is computationally expensive to evaluate, computer experiments and emulators can be used to facilitate the optimisation.

The field of Bayesian optimisation uses sequentially collected evaluations of \(g(\boldsymbol{x})\)

• place a prior distribution (eg GP) on \(g(\boldsymbol{x})\)
• collect function evaluations at points chosen sequentially via an acquisition function
• update the prior to a posterior distribution, and infer the maximum/minimum of \(g(\boldsymbol{x})\)

Uncertainty in the posterior (i.e. for \(g(\boldsymbol{x})\) at unobserved \(\boldsymbol{x}\)) leads to exploration/exploitation trade-off

The most common acquisition function is expected improvement (EI)

See Jones, Schonlau, and Welch (1998)

For a deterministic computer model and a minimisation problem, the improvement from performing one more run is given by: \[ \max(g_\min - g(\boldsymbol{x}), 0) \] where \(g_\min\) is the minimum across the model runs performed to date

This quantity is a random variable - we are uncertain about \(g(\boldsymbol{x})\) at a point we have not observed.

EI chooses \(\boldsymbol{x}\) to maximise \[ E_g\left[\max(g_\min - g(\boldsymbol{x}), 0)\,;\, \boldsymbol{g}\right] = \left[g_\min - m(\boldsymbol{x})\right]\Phi\left(\frac{g_\min - m(\boldsymbol{x})}{s(\boldsymbol{x})}\right) + s(\boldsymbol{x})\phi\left(\frac{g_\min - m(\boldsymbol{x})}{s(\boldsymbol{x})}\right) \] where \(\phi\) and \(\Phi\) are the standard normal pdf and cdf, respectively

EI is an decreasing function of \(m(\boldsymbol{x})\) and an increasing function of \(s^2(\boldsymbol{x})\), so it leads to choosing design points that either minimise the posterior mean or maximise the posterior variance

• experiment either where our uncertainty is high or near where we predict the minimum to be (explore or exploit)

A simple example: \(g(\boldsymbol{x}) = \sin(2\pi x)\) but with a different starting design using DiceOptim

xi <- matrix(c(0.1, 0.8, 0.9), ncol = 1)
fn <- function(x) sin(2 * pi * x)
y <- fn(xi)
gp <- km(design = xi, response = y, control = list(trace = F))
xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp, newdata = xs, type = "SK")

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x")
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, fn(xs), lty = 1, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)

library(DiceOptim)
xin <- max_EI(model = gp, lower = 0, upper = 1)$par

## 
## 
## Mon Sep  6 16:34:50 2021
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  1.185453e-01
##       2  1.209922e-01
##       3  1.211283e-01
##       4  1.211283e-01
##       5  1.211283e-01
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 1.211283e-01
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 6.719061e-01    G[ 1] : -1.776357e-15
## 
## Solution Found Generation 5
## Number of Generations Run 8
## 
## Mon Sep  6 16:34:51 2021
## Total run time : 0 hours 0 minutes and 1 seconds

EI(xin, gp)

## [1] 0.1211283

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
abline(v = xin)

xi <- rbind(xi, xin)
y <- c(y, fn(xin))
gp2 <- km(design = xi, response = y, control = list(trace = F))
xin <- max_EI(model = gp2, lower = 0, upper = 1, control = list(trace = F))$par

## 
## 
## Mon Sep  6 16:34:51 2021
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  9.550696e-03
##       3  5.280800e-02
##       4  5.280800e-02
##       5  5.280800e-02
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 5.280800e-02
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 7.500003e-01    G[ 1] : 6.447263e-10
## 
## Solution Found Generation 5
## Number of Generations Run 8
## 
## Mon Sep  6 16:34:51 2021
## Total run time : 0 hours 0 minutes and 0 seconds

EI(xin, gp2)

## [1] 0.052808

xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp2, newdata = xs, type = "SK")
plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp2), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
abline(v = xin)

xi <- rbind(xi, xin)
y <- c(y, fn(xin))
gp3 <- km(design = xi, response = y, control = list(trace = F))
xin <- max_EI(model = gp3, lower = 0, upper = 1, control = list(trace = F))$par

## 
## 
## Mon Sep  6 16:34:51 2021
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  2.581795e-04
##       1  5.075245e-04
##       2  5.251263e-04
##       3  5.309279e-04
##       4  5.309279e-04
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 5.309279e-04
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 4.501356e-01    G[ 1] : -6.934099e-10
## 
## Solution Found Generation 4
## Number of Generations Run 7
## 
## Mon Sep  6 16:34:51 2021
## Total run time : 0 hours 0 minutes and 0 seconds

EI(xin, gp3)

## [1] 0.0005309279

xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp3, newdata = xs, type = "SK")
plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp3), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
abline(v = xin)

Computer experiments are an important statistical contribution to the field of uncertainty quantification (UQ)

• interdisciplinary topic on the interface of Statistics and Applied Maths
• methodologies for taking account of uncertainties when mathematical and computer models are used to describe real-world phenomena

Space-filling designs and (GP) emulators are very general, and can be applied to a variety of black box computer models

• typically require a lot less knowledge about the model than alternative methods from numerical analysis (although at some loss of efficiency)

GP emulators can be used as priors for Bayesian calibration of computer models (Kennedy and O’Hagan 2001)

• eg learning tuning parameters (cf parameter estimation, albeit with various important nuances around interpretation and physical understanding)
• data fusion: combining computer model runs and data from real experiments

Now consider a more general class of models (cf preliminary material).

Let \(\boldsymbol{y}= (y_1,\ldots,y_n)^\mathrm{T}\) be iid observations from a distribution with density/mass function \(\pi(y_i\,;\,\boldsymbol{\theta},\boldsymbol{x}_i)\)

• \(\boldsymbol{\theta}\) is a \(q-\)vector of unknown parameters
• \(\boldsymbol{x}_i =(x_{1i},\ldots,x_{ki})^\mathrm{T}\) is a vector of values of \(k\) controllable variables.

The (expected) information matrix \[ M(\boldsymbol{\theta}) = E_y\left[-\frac{\partial^2l(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^\mathrm{T}}\right] \] is an important quantity for design, where \(l(\boldsymbol{\theta}) = \sum_{i=1}^n\log\pi(y_i;\,\boldsymbol{\theta},\boldsymbol{x}_i)\) (the log-likelihood).

• \(M(\boldsymbol{\theta})\) is the (asymptotic) precision for the maximum likelihood estimators \(\hat{\boldsymbol{\theta}}\).
• \(M(\boldsymbol{\theta})\) is also an asymptotic approximation to the posterior precision for \(\boldsymbol{\theta}\) in a Bayesian analysis.

Example 1: Compartmental model \[ y_i \sim N\left(c(\boldsymbol{\theta})\mu(\boldsymbol{\theta};\,x_i), \sigma^2\nu(\boldsymbol{\theta};\,x_i)\right)\,,\quad x_i\in[0,24]\,, \] with \[ \mu(\boldsymbol{\theta};\,x) = \exp(-\theta_1x)-exp(-\theta_2x)\,,\quad c(\boldsymbol{\theta}) = \frac{400\theta_2}{\theta_3(\theta_2-\theta_1)}\,,\quad \nu(\boldsymbol{\theta};\,x) = 1 + \frac{\tau^2}{\sigma^2}c(\boldsymbol{\theta})^2\mu(\boldsymbol{\theta};\,x)\,, \] for \(\theta_1, \theta_2, \theta_3, \tau^2, \sigma^2>0\).

Prior distributions (for later use):

• \(\log\theta_i\sim N(m_i, 0.05)\), with \(m_1 = \log 0.1, m_2 = 0, m_3 = \log 20\)

Ryan et al. (2014)

comp <- function(x, theta, D = 400) {
    mu <- exp(-theta[1] * x) - exp(-theta[2] * x)
    c <- (D / theta[3]) * (theta[2]) / (theta[2] - theta[1])
    c * mu }
theta <- c(.1, 1, 20)
M <- 100
par(mar = c(6, 4, 0, 1) + .1)
lapply(1:M, function(l) {
  thetat <- rlnorm(3,log(theta),rep(0.05,3))
  curve(comp(x, theta = thetat), from = 0, to = 24, ylab = "Expected concentration", 
        xlab = "Time", ylim = c(0, 20), xlim = c(0, 24), add = l!=1) })

## [[1]]
## [[1]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[1]]$y
##   [1]  0.000000  4.059900  7.167824  9.527057 11.297893 12.606716 13.553165
##   [8] 14.215783 14.656475 14.924027 15.056872 15.085282 15.033095 14.919068
##  [15] 14.757959 14.561367 14.338404 14.096218 13.840412 13.575369 13.304511
##  [22] 13.030502 12.755413 12.480840 12.208015 11.937876 11.671133 11.408315
##  [29] 11.149812 10.895904 10.646782 10.402570 10.163341  9.929125  9.699922
##  [36]  9.475706  9.256434  9.042046  8.832476  8.627645  8.427471  8.231869
##  [43]  8.040750  7.854022  7.671594  7.493376  7.319275  7.149202  6.983067
##  [50]  6.820781  6.662259  6.507413  6.356162  6.208422  6.064112  5.923154
##  [57]  5.785471  5.650986  5.519626  5.391319  5.265994  5.143581  5.024013
##  [64]  4.907224  4.793150  4.681728  4.572895  4.466592  4.362760  4.261342
##  [71]  4.162282  4.065524  3.971015  3.878703  3.788537  3.700467  3.614445
##  [78]  3.530422  3.448352  3.368191  3.289892  3.213414  3.138714  3.065750
##  [85]  2.994482  2.924871  2.856878  2.790466  2.725598  2.662237  2.600350
##  [92]  2.539901  2.480857  2.423186  2.366856  2.311835  2.258093  2.205600
##  [99]  2.154328  2.104248  2.055332
## 
## 
## [[2]]
## [[2]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[2]]$y
##   [1]  0.000000  4.393709  7.770375 10.344553 12.285965 13.728857 14.779405
##   [8] 15.521575 16.021761 16.332452 16.495136 16.542598 16.500736 16.390002
##  [15] 16.226537 16.023073 15.789644 15.534154 15.262817 14.980516 14.691076
##  [22] 14.397491 14.102095 13.806701 13.512712 13.221205 12.933002 12.648723
##  [29] 12.368828 12.093652 11.823433 11.558329 11.298440 11.043818 10.794477
##  [36] 10.550405 10.311566 10.077908  9.849366  9.625865  9.407324  9.193656
##  [43]  8.984771  8.780577  8.580979  8.385884  8.195197  8.008824  7.826673
##  [50]  7.648651  7.474667  7.304632  7.138459  6.976060  6.817352  6.662251
##  [57]  6.510676  6.362548  6.217788  6.076321  5.938071  5.802965  5.670933
##  [64]  5.541905  5.415812  5.292587  5.172166  5.054485  4.939481  4.827094
##  [71]  4.717263  4.609932  4.505042  4.402540  4.302369  4.204477  4.108813
##  [78]  4.015325  3.923965  3.834683  3.747433  3.662167  3.578842  3.497413
##  [85]  3.417836  3.340070  3.264074  3.189806  3.117229  3.046303  2.976990
##  [92]  2.909255  2.843061  2.778372  2.715156  2.653378  2.593006  2.534007
##  [99]  2.476351  2.420007  2.364945
## 
## 
## [[3]]
## [[3]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[3]]$y
##   [1]  0.000000  4.325890  7.637636 10.150523 12.034632 13.424321 14.425714
##   [8] 15.122625 15.581234 15.853789 15.981525 15.996978 15.925808 15.788241
##  [15] 15.600214 15.374270 15.120275 14.845979 14.557459 14.259476 13.955746
##  [22] 13.649164 13.341976 13.035917 12.732317 12.432189 12.136296 11.845203
##  [29] 11.559323 11.278945 11.004264 10.735400 10.472417 10.215331  9.964127
##  [36]  9.718760  9.479167  9.245268  9.016974  8.794184  8.576793  8.364695
##  [43]  8.157775  7.955923  7.759024  7.566966  7.379637  7.196925  7.018720
##  [50]  6.844916  6.675406  6.510085  6.348853  6.191609  6.038255  5.888697
##  [57]  5.742841  5.600595  5.461871  5.326583  5.194644  5.065972  4.940488
##  [64]  4.818111  4.698765  4.582374  4.468867  4.358171  4.250217  4.144937
##  [71]  4.042265  3.942136  3.844487  3.749257  3.656385  3.565814  3.477487
##  [78]  3.391348  3.307342  3.225417  3.145521  3.067605  2.991618  2.917514
##  [85]  2.845245  2.774767  2.706034  2.639004  2.573634  2.509884  2.447712
##  [92]  2.387081  2.327951  2.270287  2.214050  2.159207  2.105722  2.053562
##  [99]  2.002694  1.953086  1.904707
## 
## 
## [[4]]
## [[4]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[4]]$y
##   [1]  0.000000  3.970043  6.995035  9.280356 10.987161 12.241890 13.143738
##   [8] 13.770517 14.183258 14.429819 14.547724 14.566382 14.508836 14.393135
##  [15] 14.233406 14.040700 13.823656 13.589017 13.342043 13.086827 12.826550
##  [22] 12.563675 12.300105 12.037301 11.776382 11.518196 11.263378 11.012401
##  [29] 10.765607 10.523239 10.285460 10.052374  9.824036  9.600466  9.381655
##  [36]  9.167574  8.958175  8.753400  8.553182  8.357445  8.166111  7.979097
##  [43]  7.796319  7.617690  7.443125  7.272538  7.105842  6.942953  6.783787
##  [50]  6.628261  6.476294  6.327805  6.182717  6.040953  5.902436  5.767093
##  [57]  5.634853  5.505643  5.379395  5.256042  5.135516  5.017753  4.902691
##  [64]  4.790267  4.680421  4.573093  4.468226  4.365764  4.265652  4.167835
##  [71]  4.072261  3.978879  3.887638  3.798489  3.711385  3.626278  3.543122
##  [78]  3.461874  3.382488  3.304923  3.229137  3.155089  3.082738  3.012047
##  [85]  2.942976  2.875490  2.809551  2.745124  2.682175  2.620669  2.560574
##  [92]  2.501856  2.444485  2.388430  2.333660  2.280146  2.227859  2.176771
##  [99]  2.126855  2.078083  2.030430
## 
## 
## [[5]]
## [[5]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[5]]$y
##   [1]  0.000000  3.957814  7.042251  9.426840 11.251057 12.626993 13.644721
##   [8] 14.376614 14.880810 15.204005 15.383696 15.449982 15.427016 15.334172
##  [15] 15.186983 14.997890 14.776855 14.531843 14.269218 13.994055 13.710395
##  [22] 13.421448 13.129756 12.837327 12.545738 12.256223 11.969737 11.687018
##  [29] 11.408624 11.134971 10.866365 10.603018 10.345073 10.092615  9.845686
##  [36]  9.604289  9.368401  9.137979  8.912959  8.693268  8.478821  8.269526
##  [43]  8.065288  7.866005  7.671574  7.481892  7.296854  7.116355  6.940291
##  [50]  6.768559  6.601057  6.437685  6.278343  6.122936  5.971367  5.823544
##  [57]  5.679375  5.538770  5.401644  5.267909  5.137484  5.010286  4.886235
##  [64]  4.765255  4.647270  4.532205  4.419988  4.310549  4.203820  4.099733
##  [71]  3.998223  3.899226  3.802680  3.708524  3.616700  3.527149  3.439816
##  [78]  3.354645  3.271582  3.190577  3.111577  3.034533  2.959396  2.886121
##  [85]  2.814659  2.744967  2.677000  2.610716  2.546074  2.483032  2.421551
##  [92]  2.361592  2.303118  2.246092  2.190478  2.136241  2.083346  2.031762
##  [99]  1.981454  1.932393  1.884546
## 
## 
## [[6]]
## [[6]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[6]]$y
##   [1]  0.000000  4.382753  7.697361 10.179454 12.013251 13.342769 14.280574
##   [8] 14.914643 15.313716 15.531493 15.609910 15.581704 15.472419 15.301974
##  [15] 15.085891 14.836252 14.562454 14.271790 13.969914 13.661196 13.349005
##  [22] 13.035926 12.723935 12.414531 12.108841 11.807704 11.511731 11.221362
##  [29] 10.936898 10.658536 10.386392 10.120518  9.860921  9.607568  9.360400
##  [36]  9.119337  8.884284  8.655135  8.431775  8.214085  8.001940  7.795217
##  [43]  7.593789  7.397530  7.206315  7.020022  6.838527  6.661712  6.489458
##  [50]  6.321650  6.158174  5.998922  5.843783  5.692654  5.545431  5.402013
##  [57]  5.262303  5.126205  4.993626  4.864475  4.738664  4.616107  4.496719
##  [64]  4.380418  4.267125  4.156762  4.049253  3.944525  3.842506  3.743124
##  [71]  3.646314  3.552007  3.460139  3.370647  3.283470  3.198547  3.115821
##  [78]  3.035234  2.956732  2.880260  2.805766  2.733199  2.662508  2.593646
##  [85]  2.526565  2.461218  2.397562  2.335552  2.275146  2.216303  2.158981
##  [92]  2.103142  2.048747  1.995759  1.944141  1.893858  1.844876  1.797161
##  [99]  1.750680  1.705401  1.661293
## 
## 
## [[7]]
## [[7]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[7]]$y
##   [1]  0.000000  4.115868  7.221810  9.543665 11.257250 12.499424 13.376713
##   [8] 13.972020 14.349852 14.560396 14.642683 14.627059 14.537110 14.391157
##  [15] 14.203422 13.984937 13.744248 13.487970 13.221214 12.947917 12.671109
##  [22] 12.393110 12.115690 11.840191 11.567622 11.298735 11.034080 10.774056
##  [29] 10.518938 10.268910 10.024086  9.784522  9.550236  9.321210  9.097405
##  [36]  8.878761  8.665208  8.456663  8.253037  8.054236  7.860163  7.670719
##  [43]  7.485805  7.305319  7.129163  6.957238  6.789445  6.625688  6.465872
##  [50]  6.309905  6.157696  6.009154  5.864192  5.722725  5.584669  5.449942
##  [57]  5.318464  5.190157  5.064945  4.942753  4.823508  4.707140  4.593579
##  [64]  4.482758  4.374610  4.269071  4.166078  4.065570  3.967486  3.871769
##  [71]  3.778361  3.687206  3.598251  3.511442  3.426727  3.344055  3.263379
##  [78]  3.184648  3.107817  3.032840  2.959671  2.888268  2.818587  2.750587
##  [85]  2.684228  2.619470  2.556274  2.494603  2.434419  2.375688  2.318373
##  [92]  2.262441  2.207859  2.154593  2.102613  2.051886  2.002384  1.954075
##  [99]  1.906932  1.860927  1.816031
## 
## 
## [[8]]
## [[8]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[8]]$y
##   [1]  0.000000  4.109423  7.289744  9.733149 11.592415 12.988981 14.019373
##   [8] 14.760304 15.272736 15.605109 15.795903 15.875682 15.868716 15.794270
##  [15] 15.667630 15.500922 15.303754 15.083739 14.846901 14.597999 14.340791
##  [22] 14.078238 13.812667 13.545902 13.279369 13.014175 12.751178 12.491033
##  [29] 12.234240 11.981173 11.732107 11.487237 11.246698 11.010576 10.778919
##  [36] 10.551744 10.329046 10.110801  9.896970  9.687505  9.482348  9.281438
##  [43]  9.084706  8.892081  8.703491  8.518861  8.338116  8.161181  7.987981
##  [50]  7.818440  7.652486  7.490044  7.331042  7.175410  7.023076  6.873972
##  [57]  6.728031  6.585186  6.445371  6.308524  6.174580  6.043480  5.915162
##  [64]  5.789569  5.666641  5.546323  5.428559  5.313296  5.200480  5.090059
##  [71]  4.981982  4.876200  4.772664  4.671326  4.572140  4.475060  4.380042
##  [78]  4.287040  4.196014  4.106920  4.019718  3.934367  3.850829  3.769064
##  [85]  3.689036  3.610706  3.534040  3.459002  3.385557  3.313672  3.243313
##  [92]  3.174447  3.107044  3.041073  2.976501  2.913301  2.851443  2.790899
##  [99]  2.731640  2.673639  2.616869
## 
## 
## [[9]]
## [[9]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[9]]$y
##   [1]  0.000000  4.162507  7.346606  9.760367 11.568098 12.899544 13.857142
##   [8] 14.521757 14.957215 15.213883 15.331492 15.341379 15.268244 15.131550
##  [15] 14.946621 14.725516 14.477709 14.210643 13.930147 13.640784 13.346115
##  [22] 13.048910 12.751317 12.454990 12.161199 11.870907 11.584838 11.303527
##  [29] 11.027362 10.756614 10.491464 10.232019  9.978336  9.730425  9.488266
##  [36]  9.251810  9.020991  8.795731  8.575936  8.361507  8.152342  7.948330
##  [43]  7.749361  7.555325  7.366108  7.181599  7.001688  6.826265  6.655222
##  [50]  6.488453  6.325854  6.167321  6.012756  5.862060  5.715137  5.571894
##  [57]  5.432239  5.296082  5.163336  5.033917  4.907740  4.784726  4.664794
##  [64]  4.547868  4.433873  4.322735  4.214382  4.108745  4.005755  3.905348
##  [71]  3.807456  3.712019  3.618974  3.528261  3.439821  3.353599  3.269538
##  [78]  3.187583  3.107683  3.029786  2.953842  2.879801  2.807616  2.737240
##  [85]  2.668628  2.601736  2.536521  2.472941  2.410954  2.350521  2.291603
##  [92]  2.234161  2.178160  2.123562  2.070333  2.018438  1.967844  1.918518
##  [99]  1.870428  1.823544  1.777835
## 
## 
## [[10]]
## [[10]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[10]]$y
##   [1]  0.000000  4.423352  7.781053 10.307174 12.184850 13.557361 14.536788
##   [8] 15.210798 15.647954 15.901863 16.014434 16.018415 15.939391 15.797338
##  [15] 15.607846 15.383072 15.132486 14.863458 14.581715 14.291699 13.996846
##  [22] 13.699807 13.402620 13.106841 12.813655 12.523952 12.238395 11.957469
##  [29] 11.681522 11.410793 11.145439 10.885552 10.631175 10.382312 10.138936
##  [36]  9.901003  9.668446  9.441190  9.219150  9.002233  8.790342  8.583377
##  [43]  8.381239  8.183823  7.991029  7.802753  7.618896  7.439357  7.264038
##  [50]  7.092842  6.925674  6.762441  6.603051  6.447415  6.295444  6.147054
##  [57]  6.002160  5.860680  5.722533  5.587643  5.455932  5.327324  5.201748
##  [64]  5.079132  4.959406  4.842502  4.728353  4.616895  4.508065  4.401799
##  [71]  4.298039  4.196724  4.097798  4.001203  3.906886  3.814791  3.724868
##  [78]  3.637064  3.551330  3.467617  3.385877  3.306064  3.228132  3.152038
##  [85]  3.077737  3.005187  2.934348  2.865179  2.797640  2.731693  2.667300
##  [92]  2.604426  2.543034  2.483088  2.424556  2.367404  2.311598  2.257109
##  [99]  2.203903  2.151952  2.101226
## 
## 
## [[11]]
## [[11]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[11]]$y
##   [1]  0.000000  3.947402  6.981354  9.294389 11.038847 12.335256 13.278971
##   [8] 13.945417 14.394251 14.672646 14.817902 14.859498 14.820729 14.719992
##  [15] 14.571812 14.387649 14.176537 13.945591 13.700408 13.445387 13.183973
##  [22] 12.918864 12.652162 12.385503 12.120148 11.857070 11.597009 11.340521
##  [29] 11.088023 10.839814 10.596109 10.357049 10.122723  9.893177  9.668424
##  [36]  9.448450  9.233222  9.022693  8.816802  8.615481  8.418655  8.226246
##  [43]  8.038171  7.854346  7.674685  7.499102  7.327511  7.159826  6.995964
##  [50]  6.835839  6.679369  6.526473  6.377071  6.231084  6.088435  5.949049
##  [57]  5.812852  5.679771  5.549735  5.422675  5.298523  5.177212  5.058679
##  [64]  4.942859  4.829690  4.719112  4.611066  4.505493  4.402337  4.301543
##  [71]  4.203057  4.106825  4.012796  3.920921  3.831149  3.743432  3.657724
##  [78]  3.573977  3.492149  3.412194  3.334069  3.257733  3.183145  3.110265
##  [85]  3.039053  2.969472  2.901484  2.835052  2.770142  2.706717  2.644745
##  [92]  2.584192  2.525025  2.467213  2.410724  2.355529  2.301597  2.248901
##  [99]  2.197410  2.147099  2.097940
## 
## 
## [[12]]
## [[12]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[12]]$y
##   [1]  0.000000  4.832809  8.502644 11.263355 13.313963 14.810488 15.875230
##   [8] 16.604039 17.072022 17.338024 17.448131 17.438425 17.337146 17.166382
##  [15] 16.943400 16.681684 16.391755 16.081812 15.758231 15.425961 15.088836
##  [22] 14.749812 14.411160 14.074616 13.741495 13.412785 13.089217 12.771323
##  [29] 12.459479 12.153940 11.854867 11.562349 11.276418 10.997063 10.724241
##  [36] 10.457883 10.197902  9.944198  9.696658  9.455165  9.219596  8.989826
##  [43]  8.765726  8.547169  8.334027  8.126174  7.923484  7.725833  7.533099
##  [50]  7.345163  7.161908  6.983219  6.808983  6.639091  6.473434  6.311909
##  [57]  6.154412  6.000844  5.851106  5.705104  5.562745  5.423937  5.288592
##  [64]  5.156625  5.027950  4.902485  4.780152  4.660871  4.544566  4.431163
##  [71]  4.320590  4.212777  4.107653  4.005153  3.905210  3.807762  3.712745
##  [78]  3.620099  3.529764  3.441684  3.355802  3.272063  3.190414  3.110802
##  [85]  3.033176  2.957488  2.883688  2.811730  2.741567  2.673155  2.606451
##  [92]  2.541411  2.477993  2.416159  2.355867  2.297080  2.239760  2.183870
##  [99]  2.129374  2.076239  2.024430
## 
## 
## [[13]]
## [[13]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[13]]$y
##   [1]  0.000000  4.397431  7.701479 10.159551 11.963619 13.262627 14.172123
##   [8] 14.781732 15.160961 15.363702 15.431729 15.397409 15.285813 15.116345
##  [15] 14.904015 14.660422 14.394521 14.113213 13.821807 13.524378 13.224045
##  [22] 12.923182 12.623593 12.326633 12.033314 11.744383 11.460379 11.181684
##  [29] 10.908557 10.641164 10.379597 10.123895  9.874054  9.630039  9.391790
##  [36]  9.159230  8.932270  8.710811  8.494746  8.283967  8.078360  7.877811
##  [43]  7.682206  7.491431  7.305373  7.123919  6.946960  6.774387  6.606093
##  [50]  6.441974  6.281928  6.125854  5.973656  5.825236  5.680503  5.539364
##  [57]  5.401731  5.267517  5.136637  5.009009  4.884551  4.763185  4.644835
##  [64]  4.529425  4.416883  4.307137  4.200118  4.095757  3.993990  3.894751
##  [71]  3.797978  3.703610  3.611586  3.521849  3.434341  3.349008  3.265795
##  [78]  3.184650  3.105521  3.028358  2.953112  2.879736  2.808183  2.738408
##  [85]  2.670366  2.604016  2.539314  2.476219  2.414692  2.354694  2.296187
##  [92]  2.239134  2.183498  2.129244  2.076339  2.024748  1.974439  1.925380
##  [99]  1.877540  1.830889  1.785396
## 
## 
## [[14]]
## [[14]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[14]]$y
##   [1]  0.000000  3.993248  7.022773  9.301015 10.994006 12.231497 13.114879
##   [8] 13.723369 14.118853 14.349668 14.453560 14.459997 14.391974 14.267428
##  [15] 14.100344 13.901616 13.679724 13.441260 13.191344 12.933941 12.672120
##  [22] 12.408245 12.144133 11.881171 11.620415 11.362658 11.108492 10.858350
##  [29] 10.612544 10.371289 10.134726  9.902940  9.675971  9.453826  9.236485
##  [36]  9.023911  8.816048  8.612832  8.414189  8.220042  8.030308  7.844900
##  [43]  7.663731  7.486714  7.313761  7.144784  6.979695  6.818409  6.660840
##  [50]  6.506905  6.356522  6.209610  6.066090  5.925885  5.788918  5.655114
##  [57]  5.524403  5.396711  5.271970  5.150112  5.031070  4.914779  4.801176
##  [64]  4.690199  4.581786  4.475880  4.372421  4.271354  4.172622  4.076173
##  [71]  3.981953  3.889911  3.799997  3.712160  3.626355  3.542532  3.460647
##  [78]  3.380655  3.302512  3.226175  3.151602  3.078753  3.007589  2.938069
##  [85]  2.870156  2.803813  2.739003  2.675691  2.613843  2.553425  2.494403
##  [92]  2.436745  2.380420  2.325397  2.271646  2.219137  2.167842  2.117733
##  [99]  2.068782  2.020962  1.974248
## 
## 
## [[15]]
## [[15]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[15]]$y
##   [1]  0.000000  3.705685  6.601426  8.847428 10.572558 11.880465 12.854509
##   [8] 13.561726 14.056012 14.380697 14.570608 14.653726 14.652532 14.585076
##  [15] 14.465846 14.306465 14.116249 13.902662 13.671676 13.428062 13.175630
##  [22] 12.917413 12.655821 12.392764 12.129749 11.867963 11.608329 11.351567
##  [29] 11.098227 10.848727 10.603377 10.362404 10.125964  9.894160  9.667053
##  [36]  9.444668  9.227004  9.014039  8.805734  8.602038  8.402887  8.208215
##  [43]  8.017945  7.832000  7.650297  7.472755  7.299288  7.129811  6.964241
##  [50]  6.802492  6.644480  6.490124  6.339342  6.192053  6.048178  5.907640
##  [57]  5.770362  5.636271  5.505292  5.377354  5.252387  5.130322  5.011093
##  [64]  4.894634  4.780880  4.669769  4.561240  4.455233  4.351689  4.250551
##  [71]  4.151764  4.055272  3.961023  3.868964  3.779045  3.691215  3.605426
##  [78]  3.521632  3.439784  3.359839  3.281752  3.205480  3.130980  3.058212
##  [85]  2.987135  2.917710  2.849899  2.783663  2.718967  2.655775  2.594051
##  [92]  2.533762  2.474874  2.417354  2.361172  2.306295  2.252694  2.200338
##  [99]  2.149199  2.099249  2.050459
## 
## 
## [[16]]
## [[16]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[16]]$y
##   [1]  0.000000  4.355182  7.648845 10.117198 11.944365 13.273863 14.217542
##   [8] 14.862564 15.276845 15.513287 15.613094 15.608346 15.524012 15.379513
##  [15] 15.189951 14.967056 14.719933 14.455640 14.179639 13.896152 13.608431
##  [22] 13.318973 13.029689 12.742035 12.457109 12.175734 11.898517 11.625901
##  [29] 11.358197 11.095619 10.838301 10.586319 10.339704 10.098450  9.862526
##  [36]  9.631881  9.406448  9.186149  8.970899  8.760607  8.555177  8.354512
##  [43]  8.158513  7.967080  7.780114  7.597517  7.419189  7.245036  7.074962
##  [50]  6.908872  6.746677  6.588284  6.433607  6.282559  6.135055  5.991012
##  [57]  5.850350  5.712990  5.578854  5.447867  5.319955  5.195046  5.073069
##  [64]  4.953956  4.837639  4.724054  4.613135  4.504821  4.399049  4.295761
##  [71]  4.194899  4.096404  4.000222  3.906298  3.814580  3.725015  3.637553
##  [78]  3.552145  3.468741  3.387297  3.307764  3.230099  3.154257  3.080197
##  [85]  3.007875  2.937251  2.868285  2.800939  2.735174  2.670953  2.608240
##  [92]  2.546999  2.487197  2.428798  2.371771  2.316083  2.261702  2.208598
##  [99]  2.156741  2.106101  2.056651
## 
## 
## [[17]]
## [[17]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[17]]$y
##   [1]  0.000000  4.331234  7.645833 10.161244 12.048844 13.443693 14.452221
##   [8] 15.158283 15.627943 15.913238 16.055144 16.085924 16.030966 15.910246
##  [15] 15.739467 15.530970 15.294443 15.037485 14.766048 14.484788 14.197337
##  [22] 13.906524 13.614543 13.323092 13.033473 12.746682 12.463471 12.184403
##  [29] 11.909889 11.640226 11.375617 11.116196 10.862037 10.613175 10.369609
##  [36] 10.131313  9.898241  9.670332  9.447515  9.229708  9.016825  8.808775
##  [43]  8.605466  8.406801  8.212685  8.023022  7.837715  7.656670  7.479792
##  [50]  7.306989  7.138169  6.973243  6.812121  6.654718  6.500949  6.350729
##  [57]  6.203979  6.060618  5.920569  5.783754  5.650101  5.519535  5.391986
##  [64]  5.267384  5.145661  5.026751  4.910588  4.797110  4.686254  4.577959
##  [71]  4.472167  4.368820  4.267861  4.169235  4.072888  3.978768  3.886822
##  [78]  3.797001  3.709256  3.623539  3.539802  3.458001  3.378090  3.300025
##  [85]  3.223765  3.149267  3.076490  3.005396  2.935944  2.868097  2.801818
##  [92]  2.737071  2.673819  2.612030  2.551668  2.492702  2.435098  2.378825
##  [99]  2.323853  2.270151  2.217690
## 
## 
## [[18]]
## [[18]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[18]]$y
##   [1]  0.000000  3.956527  7.007357  9.339776 11.102782 12.414908 13.370435
##   [8] 14.044346 14.496268 14.773603 14.914027 14.947475 14.897725 14.783652
##  [15] 14.620232 14.419339 14.190380 13.940795 13.676468 13.402035 13.121150
##  [22] 12.836679 12.550867 12.265461 11.981815 11.700973 11.423729 11.150682
##  [29] 10.882276 10.618832 10.360572 10.107645  9.860139  9.618092  9.381510
##  [36]  9.150367  8.924617  8.704195  8.489027  8.279024  8.074096  7.874143
##  [43]  7.679066  7.488760  7.303122  7.122047  6.945430  6.773169  6.605161
##  [50]  6.441305  6.281501  6.125652  5.973662  5.825437  5.680885  5.539916
##  [57]  5.402442  5.268376  5.137636  5.010138  4.885804  4.764554  4.646312
##  [64]  4.531004  4.418557  4.308900  4.201964  4.097682  3.995988  3.896817
##  [71]  3.800107  3.705798  3.613829  3.524142  3.436681  3.351390  3.268216
##  [78]  3.187107  3.108010  3.030876  2.955657  2.882304  2.810772  2.741015
##  [85]  2.672989  2.606652  2.541961  2.478875  2.417355  2.357362  2.298857
##  [92]  2.241805  2.186168  2.131912  2.079003  2.027407  1.977091  1.928024
##  [99]  1.880175  1.833514  1.788010
## 
## 
## [[19]]
## [[19]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[19]]$y
##   [1]  0.000000  3.947885  6.997648  9.334013 11.104140 12.425251 13.390710
##   [8] 14.074869 14.536932 14.824028 14.973670 15.015701 14.973861 14.867022
##  [15] 14.710177 14.515233 14.291636 14.046871 13.786864 13.516297 13.238863
##  [22] 12.957466 12.674386 12.391400 12.109894 11.830934 11.555338 11.283726
##  [29] 11.016558 10.754172 10.496806 10.244620  9.997712  9.756134  9.519898
##  [36]  9.288987  9.063364  8.842970  8.627736  8.417582  8.212419  8.012157
##  [43]  7.816697  7.625942  7.439790  7.258142  7.080897  6.907954  6.739214
##  [50]  6.574580  6.413955  6.257243  6.104352  5.955190  5.809668  5.667697
##  [57]  5.529193  5.394070  5.262248  5.133645  5.008184  4.885788  4.766382
##  [64]  4.649894  4.536252  4.425388  4.317232  4.211719  4.108785  4.008366
##  [71]  3.910402  3.814831  3.721597  3.630641  3.541907  3.455343  3.370894
##  [78]  3.288509  3.208137  3.129730  3.053239  2.978617  2.905819  2.834800
##  [85]  2.765517  2.697928  2.631990  2.567664  2.504910  2.443689  2.383965
##  [92]  2.325701  2.268860  2.213409  2.159313  2.106539  2.055054  2.004829
##  [99]  1.955830  1.908029  1.861397
## 
## 
## [[20]]
## [[20]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[20]]$y
##   [1]  0.000000  4.691212  8.253275 10.935385 12.932192 14.395733 15.444748
##   [8] 16.171966 16.649790 16.934748 17.070962 17.092860 17.027303 16.895234
##  [15] 16.712980 16.493254 16.245953 15.978773 15.697689 15.407333 15.111291
##  [22] 14.812329 14.512575 14.213659 13.916823 13.623004 13.332907 13.047048
##  [29] 12.765806 12.489444 12.218143 11.952015 11.691120 11.435480 11.185087
##  [36] 10.939909 10.699895 10.464982 10.235100 10.010167  9.790100  9.574809
##  [43]  9.364205  9.158196  8.956691  8.759596  8.566821  8.378274  8.193866
##  [50]  8.013508  7.837114  7.664597  7.495874  7.330862  7.169480  7.011649
##  [57]  6.857291  6.706329  6.558691  6.414301  6.273090  6.134987  5.999924
##  [64]  5.867835  5.738653  5.612315  5.488758  5.367921  5.249745  5.134170
##  [71]  5.021139  4.910597  4.802488  4.696760  4.593359  4.492234  4.393336
##  [78]  4.296615  4.202023  4.109514  4.019042  3.930561  3.844028  3.759400
##  [85]  3.676635  3.595693  3.516532  3.439114  3.363401  3.289354  3.216938
##  [92]  3.146116  3.076853  3.009115  2.942868  2.878079  2.814717  2.752750
##  [99]  2.692147  2.632879  2.574915
## 
## 
## [[21]]
## [[21]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[21]]$y
##   [1]  0.000000  4.301897  7.606123 10.122880 12.018531 13.424730 14.445660
##   [8] 15.163749 15.644205 15.938604 16.087730 16.123826 16.072371 15.953492
##  [15] 15.783078 15.573666 15.335139 15.075280 14.800211 14.514736 14.222625
##  [22] 13.926820 13.629616 13.332793 13.037726 12.745469 12.456821 12.172385
##  [29] 11.892604 11.617799 11.348192 11.083931 10.825103 10.571748 10.323871
##  [36] 10.081449  9.844436  9.612771  9.386381  9.165182  8.949085  8.737995
##  [43]  8.531814  8.330443  8.133782  7.941728  7.754182  7.571044  7.392213
##  [50]  7.217593  7.047087  6.880601  6.718041  6.559316  6.404337  6.253017
##  [57]  6.105269  5.961011  5.820159  5.682634  5.548358  5.417253  5.289246
##  [64]  5.164263  5.042233  4.923086  4.806755  4.693172  4.582273  4.473994
##  [71]  4.368274  4.265052  4.164269  4.065867  3.969791  3.875985  3.784395
##  [78]  3.694970  3.607657  3.522408  3.439174  3.357906  3.278559  3.201086
##  [85]  3.125444  3.051590  2.979481  2.909076  2.840334  2.773217  2.707686
##  [92]  2.643703  2.581232  2.520238  2.460684  2.402538  2.345766  2.290336
##  [99]  2.236215  2.183373  2.131780
## 
## 
## [[22]]
## [[22]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[22]]$y
##   [1]  0.000000  4.147804  7.323833  9.734713 11.543594 12.879282 13.843462
##   [8] 14.516395 14.961425 15.228532 15.357142 15.378349 15.316662 15.191395
##  [15] 15.017758 14.807717 14.570684 14.314047 14.043603 13.763888 13.478447
##  [22] 13.190041 12.900813 12.612418 12.326130 12.042917 11.763513 11.488462
##  [29] 11.218161 10.952894 10.692852 10.438158 10.188878  9.945036  9.706623
##  [36]  9.473604  9.245925  9.023518  8.806302  8.594191  8.387089  8.184900
##  [43]  7.987524  7.794859  7.606803  7.423253  7.244109  7.069268  6.898633
##  [50]  6.732105  6.569587  6.410985  6.256206  6.105159  5.957756  5.813908
##  [57]  5.673532  5.536543  5.402860  5.272404  5.145097  5.020864  4.899629
##  [64]  4.781322  4.665870  4.553207  4.443263  4.335974  4.231275  4.129105
##  [71]  4.029401  3.932105  3.837159  3.744504  3.654087  3.565854  3.479751
##  [78]  3.395726  3.313731  3.233716  3.155633  3.079435  3.005077  2.932515
##  [85]  2.861704  2.792604  2.725172  2.659368  2.595154  2.532490  2.471338
##  [92]  2.411664  2.353431  2.296603  2.241148  2.187032  2.134223  2.082688
##  [99]  2.032398  1.983323  1.935432
## 
## 
## [[23]]
## [[23]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[23]]$y
##   [1]  0.000000  3.762801  6.705776  8.990424 10.746787 12.079570 13.073069
##   [8] 13.795140 14.300401 14.632804 14.827711 14.913567 14.913245 14.845130
##  [15] 14.723994 14.561699 14.367764 14.149821 13.913983 13.665141 13.407201
##  [22] 13.143278 12.875849 12.606880 12.337921 12.070196 11.804659 11.542052
##  [29] 11.282944 11.027769 10.776848 10.530416 10.288637 10.051617  9.819421
##  [36]  9.592076  9.369581  9.151915  8.939038  8.730896  8.527427  8.328560
##  [43]  8.134218  7.944320  7.758782  7.577519  7.400443  7.227466  7.058502
##  [50]  6.893463  6.732263  6.574817  6.421040  6.270849  6.124162  5.980901
##  [57]  5.840985  5.704337  5.570883  5.440549  5.313261  5.188949  5.067545
##  [64]  4.948980  4.833187  4.720104  4.609665  4.501810  4.396478  4.293610
##  [71]  4.193149  4.095039  3.999223  3.905650  3.814265  3.725019  3.637861
##  [78]  3.552742  3.469615  3.388433  3.309150  3.231723  3.156106  3.082260
##  [85]  3.010141  2.939709  2.870926  2.803751  2.738149  2.674081  2.611513
##  [92]  2.550409  2.490734  2.432455  2.375541  2.319957  2.265675  2.212662
##  [99]  2.160890  2.110330  2.060952
## 
## 
## [[24]]
## [[24]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[24]]$y
##   [1]  0.000000  3.885404  6.925202  9.286109 11.102328 12.481878 13.511700
##   [8] 14.261757 14.788347 15.136761 15.343429 15.437647 15.442968 15.378319
##  [15] 15.258903 15.096929 14.902191 14.682542 14.444271 14.192410 13.930978
##  [22] 13.663183 13.391573 13.118172 12.844582 12.572061 12.301597 12.033957
##  [29] 11.769732 11.509371 11.253212 11.001500 10.754409 10.512056 10.274511
##  [36] 10.041807  9.813951  9.590924  9.372690  9.159202  8.950398  8.746211
##  [43]  8.546567  8.351387  8.160590  7.974091  7.791806  7.613649  7.439534
##  [50]  7.269376  7.103089  6.940589  6.781794  6.626621  6.474991  6.326822
##  [57]  6.182039  6.040565  5.902325  5.767245  5.635254  5.506283  5.380261
##  [64]  5.257123  5.136802  5.019234  4.904356  4.792107  4.682426  4.575256
##  [71]  4.470538  4.368217  4.268237  4.170546  4.075090  3.981820  3.890684
##  [78]  3.801633  3.714621  3.629601  3.546526  3.465353  3.386037  3.308537
##  [85]  3.232811  3.158818  3.086519  3.015874  2.946847  2.879399  2.813495
##  [92]  2.749099  2.686177  2.624696  2.564621  2.505922  2.448566  2.392523
##  [99]  2.337763  2.284256  2.231973
## 
## 
## [[25]]
## [[25]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[25]]$y
##   [1]  0.000000  4.339988  7.590228  9.999871 11.761652 13.024655 13.904184
##   [8] 14.489401 14.849247 15.037009 15.093868 15.051634 14.934867 14.762521
##  [15] 14.549209 14.306191 14.042128 13.763676 13.475936 13.182813 12.887279
##  [22] 12.591595 12.297464 12.006165 11.718646 11.435600 11.157526 10.884773
##  [29] 10.617572 10.356068 10.100337  9.850406  9.606260  9.367856  9.135130
##  [36]  8.908001  8.686375  8.470153  8.259228  8.053488  7.852823  7.657117
##  [43]  7.466259  7.280133  7.098630  6.921637  6.749046  6.580750  6.416644
##  [50]  6.256626  6.100594  5.948450  5.800098  5.655444  5.514397  5.376866
##  [57]  5.242764  5.112006  4.984509  4.860192  4.738974  4.620780  4.505533
##  [64]  4.393161  4.283591  4.176754  4.072581  3.971007  3.871966  3.775395
##  [71]  3.681232  3.589419  3.499895  3.412603  3.327489  3.244498  3.163577
##  [78]  3.084674  3.007739  2.932722  2.859577  2.788256  2.718714  2.650906
##  [85]  2.584790  2.520322  2.457462  2.396171  2.336408  2.278135  2.221316
##  [92]  2.165914  2.111893  2.059220  2.007861  1.957783  1.908954  1.861342
##  [99]  1.814918  1.769652  1.725515
## 
## 
## [[26]]
## [[26]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[26]]$y
##   [1]  0.000000  4.143794  7.276263  9.622757 11.358866 12.621400 13.516943
##   [8] 14.128519 14.520792 14.744116 14.837691 14.832026 14.750857 14.612640
##  [15] 14.431718 14.219231 13.983820 13.732181 13.469493 13.199758 12.926057
##  [22] 12.650755 12.375661 12.102152 11.831265 11.563779 11.300266 11.041144
##  [29] 10.786706 10.537151 10.292607 10.053144  9.818788  9.589535  9.365353
##  [36]  9.146192  8.931990  8.722670  8.518152  8.318349  8.123169  7.932520
##  [43]  7.746306  7.564435  7.386810  7.213338  7.043925  6.878480  6.716913
##  [50]  6.559134  6.405056  6.254593  6.107662  5.964180  5.824066  5.687243
##  [57]  5.553633  5.423161  5.295753  5.171338  5.049846  4.931207  4.815356
##  [64]  4.702226  4.591753  4.483876  4.378534  4.275666  4.175214  4.077123
##  [71]  3.981336  3.887800  3.796461  3.707268  3.620170  3.535119  3.452066
##  [78]  3.370964  3.291767  3.214431  3.138912  3.065167  2.993155  2.922835
##  [85]  2.854166  2.787111  2.721631  2.657690  2.595251  2.534279  2.474739
##  [92]  2.416598  2.359823  2.304382  2.250243  2.197377  2.145752  2.095340
##  [99]  2.046113  1.998042  1.951100
## 
## 
## [[27]]
## [[27]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[27]]$y
##   [1]  0.000000  4.287338  7.532565  9.966472 11.769224 13.081454 14.012925
##   [8] 14.649298 15.057409 15.289401 15.385939 15.378730 15.292484 15.146450
##  [15] 14.955614 14.731633 14.483564 14.218439 13.941706 13.657580 13.369309
##  [22] 13.079394 12.789747 12.501826 12.216733 11.935291 11.658110 11.385633
##  [29] 11.118171 10.855936 10.599062 10.347622 10.101646  9.861124  9.626024
##  [36]  9.396289  9.171851  8.952627  8.738530  8.529464  8.325330  8.126028
##  [43]  7.931455  7.741507  7.556083  7.375080  7.198397  7.025934  6.857594
##  [50]  6.693279  6.532896  6.376352  6.223555  6.074417  5.928850  5.786770
##  [57]  5.648094  5.512740  5.380629  5.251683  5.125826  5.002986  4.883089
##  [64]  4.766065  4.651846  4.540364  4.431553  4.325350  4.221692  4.120518
##  [71]  4.021769  3.925386  3.831313  3.739494  3.649876  3.562406  3.477032
##  [78]  3.393704  3.312373  3.232991  3.155511  3.079888  3.006078  2.934036
##  [85]  2.863721  2.795091  2.728106  2.662726  2.598913  2.536630  2.475838
##  [92]  2.416504  2.358592  2.302068  2.246898  2.193050  2.140493  2.089196
##  [99]  2.039127  1.990259  1.942562
## 
## 
## [[28]]
## [[28]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[28]]$y
##   [1]  0.000000  3.966416  7.054097  9.439493 11.264002 12.640922 13.661031
##   [8] 14.397050 14.907230 15.238221 15.427377 15.504600 15.493819 15.414175
##  [15] 15.280975 15.106449 14.900365 14.670516 14.423115 14.163106 13.894419
##  [22] 13.620173 13.342832 13.064344 12.786236 12.509705 12.235679 11.964874
##  [29] 11.697835 11.434973 11.176589 10.922900 10.674052 10.430139 10.191212
##  [36]  9.957286  9.728351  9.504377  9.285317  9.071111  8.861691  8.656981
##  [43]  8.456899  8.261361  8.070280  7.883568  7.701133  7.522887  7.348740
##  [50]  7.178603  7.012388  6.850008  6.691377  6.536411  6.385026  6.237142
##  [57]  6.092679  5.951558  5.813703  5.679038  5.547491  5.418990  5.293464
##  [64]  5.170845  5.051065  4.934059  4.819763  4.708115  4.599052  4.492516
##  [71]  4.388447  4.286789  4.187486  4.090483  3.995727  3.903165  3.812748
##  [78]  3.724426  3.638149  3.553871  3.471545  3.391127  3.312571  3.235835
##  [85]  3.160877  3.087654  3.016129  2.946260  2.878009  2.811340  2.746215
##  [92]  2.682598  2.620456  2.559753  2.500456  2.442532  2.385951  2.330680
##  [99]  2.276690  2.223950  2.172432
## 
## 
## [[29]]
## [[29]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[29]]$y
##   [1]  0.000000  4.087000  7.232863  9.633366 11.444023 12.788361 13.764483
##   [8] 14.450289 14.907630 15.185602 15.323164 15.351226 15.294298 15.171806
##  [15] 14.999141 14.788484 14.549468 14.289704 14.015193 13.730664 13.439830
##  [22] 13.145600 12.850248 12.555539 12.262840 11.973197 11.687407 11.406067
##  [29] 11.129617 10.858373 10.592553 10.332297 10.077687  9.828757  9.585502
##  [36]  9.347893  9.115876  8.889380  8.668323  8.452614  8.242154  8.036840
##  [43]  7.836565  7.641222  7.450700  7.264891  7.083685  6.906976  6.734656
##  [50]  6.566620  6.402765  6.242989  6.087193  5.935278  5.787150  5.642715
##  [57]  5.501882  5.364561  5.230666  5.100112  4.972814  4.848693  4.727670
##  [64]  4.609666  4.494607  4.382420  4.273033  4.166376  4.062381  3.960982
##  [71]  3.862113  3.765712  3.671718  3.580069  3.490708  3.403577  3.318622
##  [78]  3.235787  3.155019  3.076267  2.999482  2.924612  2.851612  2.780433
##  [85]  2.711032  2.643362  2.577382  2.513049  2.450321  2.389159  2.329524
##  [92]  2.271377  2.214682  2.159402  2.105502  2.052947  2.001704  1.951740
##  [99]  1.903023  1.855522  1.809207
## 
## 
## [[30]]
## [[30]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[30]]$y
##   [1]  0.000000  4.167612  7.340211  9.734258 11.519568 12.829358 13.768119
##   [8] 14.417803 14.842674 15.093118 15.208627 15.220149 15.151927 15.022940
##  [15] 14.848042 14.638847 14.404428 14.151868 13.886683 13.613167 13.334650
##  [22] 13.053713 12.772342 12.492064 12.214041 11.939155 11.668064 11.401253
##  [29] 11.139074 10.881770 10.629506 10.382380 10.140444  9.903708  9.672157
##  [36]  9.445750  9.224433  9.008135  8.796780  8.590282  8.388551  8.191494
##  [43]  7.999018  7.811025  7.627420  7.448108  7.272992  7.101979  6.934975
##  [50]  6.771889  6.612632  6.457114  6.305249  6.156953  6.012142  5.870735
##  [57]  5.732651  5.597815  5.466149  5.337579  5.212032  5.089438  4.969728
##  [64]  4.852832  4.738686  4.627225  4.518385  4.412105  4.308326  4.206987
##  [71]  4.108031  4.011403  3.917048  3.824913  3.734944  3.647092  3.561306
##  [78]  3.477538  3.395740  3.315867  3.237872  3.161712  3.087343  3.014723
##  [85]  2.943812  2.874568  2.806953  2.740929  2.676458  2.613503  2.552029
##  [92]  2.492000  2.433384  2.376147  2.320256  2.265679  2.212387  2.160348
##  [99]  2.109532  2.059913  2.011460
## 
## 
## [[31]]
## [[31]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[31]]$y
##   [1]  0.000000  4.041152  7.176949  9.590910 11.429768 12.810831 13.827873
##   [8] 14.555848 15.054657 15.372162 15.546599 15.608502 15.582250 15.487296
##  [15] 15.339156 15.150201 14.930283 14.687240 14.427305 14.155422 13.875512
##  [22] 13.590672 13.303342 13.015444 12.728476 12.443606 12.161735 11.883552
##  [29] 11.609579 11.340201 11.075698 10.816265 10.562030 10.313068 10.069413
##  [36]  9.831065  9.598000  9.370173  9.147524  8.929982  8.717466  8.509891
##  [43]  8.307164  8.109191  7.915876  7.727122  7.542830  7.362903  7.187243
##  [50]  7.015754  6.848341  6.684911  6.525370  6.369629  6.217599  6.069192
##  [57]  5.924323  5.782909  5.644868  5.510120  5.378587  5.250193  5.124862
##  [64]  5.002522  4.883102  4.766532  4.652745  4.541673  4.433253  4.327421
##  [71]  4.224115  4.123275  4.024842  3.928759  3.834970  3.743420  3.654055
##  [78]  3.566823  3.481674  3.398558  3.317425  3.238230  3.160925  3.085466
##  [85]  3.011808  2.939908  2.869725  2.801217  2.734345  2.669069  2.605351
##  [92]  2.543155  2.482443  2.423181  2.365333  2.308867  2.253748  2.199945
##  [99]  2.147427  2.096162  2.046121
## 
## 
## [[32]]
## [[32]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[32]]$y
##   [1]  0.000000  4.059946  7.201612  9.612030 11.440606 12.806673 13.805535
##   [8] 14.513295 14.990710 15.286277 15.438698 15.478844 15.431335 15.315792
##  [15] 15.147845 14.939935 14.701955 14.441764 14.165598 13.878395 13.584055
##  [22] 13.285655 12.985611 12.685815 12.387736 12.092512 11.801015 11.513904
##  [29] 11.231671 10.954673 10.683163 10.417310 10.157216  9.902931  9.654465
##  [36]  9.411797  9.174879  8.943645  8.718017  8.497903  8.283204  8.073816
##  [43]  7.869630  7.670535  7.476419  7.287170  7.102673  6.922818  6.747494
##  [50]  6.576591  6.410002  6.247620  6.089342  5.935066  5.784693  5.638125
##  [57]  5.495267  5.356025  5.220308  5.088029  4.959100  4.833437  4.710957
##  [64]  4.591579  4.475227  4.361822  4.251290  4.143559  4.038558  3.936217
##  [71]  3.836470  3.739250  3.644494  3.552139  3.462124  3.374390  3.288879
##  [78]  3.205536  3.124304  3.045131  2.967964  2.892752  2.819447  2.747999
##  [85]  2.678361  2.610489  2.544336  2.479860  2.417017  2.355767  2.296070
##  [92]  2.237885  2.181174  2.125901  2.072028  2.019520  1.968344  1.918463
##  [99]  1.869847  1.822463  1.776280
## 
## 
## [[33]]
## [[33]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[33]]$y
##   [1]  0.000000  3.962634  7.064007  9.473717 11.328324 12.737777 13.790594
##   [8] 14.558030 15.097437 15.454969 15.667760 15.765682 15.772756 15.708293
##  [15] 15.587810 15.423768 15.226169 15.003037 14.760797 14.504593 14.238536
##  [22] 13.965904 13.689309 13.410821 13.132081 12.854380 12.578729 12.305917
##  [29] 12.036549 11.771088 11.509882 11.253184 11.001175 10.753974 10.511656
##  [36] 10.274257 10.041784  9.814219  9.591530  9.373666  9.160567  8.952165
##  [43]  8.748385  8.549148  8.354372  8.163971  7.977859  7.795950  7.618156
##  [50]  7.444391  7.274568  7.108602  6.946409  6.787906  6.633010  6.481642
##  [57]  6.333723  6.189175  6.047921  5.909889  5.775004  5.643196  5.514395
##  [64]  5.388533  5.265542  5.145357  5.027915  4.913153  4.801010  4.691427
##  [71]  4.584344  4.479705  4.377455  4.277538  4.179902  4.084494  3.991264
##  [78]  3.900162  3.811139  3.724148  3.639143  3.556078  3.474909  3.395593
##  [85]  3.318087  3.242350  3.168342  3.096023  3.025355  2.956300  2.888821
##  [92]  2.822883  2.758449  2.695486  2.633961  2.573839  2.515090  2.457682
##  [99]  2.401584  2.346767  2.293201
## 
## 
## [[34]]
## [[34]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[34]]$y
##   [1]  0.000000  4.136106  7.279025  9.645462 11.405315 12.691778 13.609253
##   [8] 14.239569 14.646848 14.881326 14.982353 14.980737 14.900596 14.760792
##  [15] 14.576075 14.357966 14.115456 13.855553 13.583711 13.304166 13.020199
##  [22] 12.734345 12.448552 12.164312 11.882757 11.604739 11.330891 11.061678
##  [29] 10.797428 10.538368 10.284643 10.036337  9.793484  9.556084  9.324106
##  [36]  9.097498  8.876193  8.660111  8.449164  8.243256  8.042289  7.846161
##  [43]  7.654768  7.468007  7.285773  7.107963  6.934474  6.765206  6.600058
##  [50]  6.438933  6.281735  6.128370  5.978744  5.832768  5.690354  5.551415
##  [57]  5.415867  5.283627  5.154616  5.028753  4.905963  4.786171  4.669304
##  [64]  4.555290  4.444060  4.335545  4.229680  4.126400  4.025642  3.927344
##  [71]  3.831446  3.737890  3.646618  3.557575  3.470706  3.385958  3.303280
##  [78]  3.222621  3.143931  3.067162  2.992268  2.919203  2.847921  2.778381
##  [85]  2.710538  2.644352  2.579782  2.516789  2.455334  2.395380  2.336889
##  [92]  2.279827  2.224158  2.169849  2.116865  2.065175  2.014748  1.965552
##  [99]  1.917557  1.870734  1.825054
## 
## 
## [[35]]
## [[35]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[35]]$y
##   [1]  0.000000  3.917240  6.920608  9.204346 10.921802 12.194032 13.116593
##   [8] 13.764902 14.198475 14.464271 14.599333 14.632874 14.587924 14.482633
##  [15] 14.331295 14.145162 13.933085 13.702018 13.457416 13.203556 12.943782
##  [22] 12.680702 12.416345 12.152282 11.889724 11.629599 11.372609 11.119282
##  [29] 10.870008 10.625067 10.384655 10.148901  9.917882  9.691634  9.470163
##  [36]  9.253449  9.041454  8.834127  8.631405  8.433216  8.239485  8.050131
##  [43]  7.865070  7.684217  7.507487  7.334793  7.166049  7.001169  6.840068
##  [50]  6.682664  6.528872  6.378614  6.231807  6.088376  5.948242  5.811330
##  [57]  5.677568  5.546884  5.419205  5.294465  5.172595  5.053530  4.937205
##  [64]  4.823557  4.712525  4.604048  4.498069  4.394528  4.293371  4.194542
##  [71]  4.097988  4.003657  3.911497  3.821458  3.733492  3.647551  3.563588
##  [78]  3.481558  3.401416  3.323119  3.246624  3.171890  3.098876  3.027543
##  [85]  2.957852  2.889765  2.823246  2.758258  2.694765  2.632734  2.572131
##  [92]  2.512924  2.455079  2.398565  2.343352  2.289411  2.236711  2.185224
##  [99]  2.134922  2.085779  2.037766
## 
## 
## [[36]]
## [[36]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[36]]$y
##   [1]  0.000000  4.003958  7.065259  9.384820 11.121203 12.399532 13.318535
##   [8] 13.956100 14.373662 14.619670 14.732323 14.741726 14.671595 14.540605
##  [15] 14.363454 14.151696 13.914408 13.658713 13.390189 13.113199 12.831145
##  [22] 12.546673 12.261833 11.978204 11.696996 11.419128 11.145289 10.875990
##  [29] 10.611598 10.352371 10.098482  9.850033  9.607077  9.369624  9.137655
##  [36]  8.911123  8.689967  8.474109  8.263463  8.057935  7.857426  7.661832
##  [43]  7.471049  7.284970  7.103490  6.926502  6.753901  6.585584  6.421447
##  [50]  6.261389  6.105313  5.953120  5.804715  5.660006  5.518901  5.381310
##  [57]  5.247148  5.116329  4.988771  4.864391  4.743112  4.624855  4.509547
##  [64]  4.397113  4.287482  4.180584  4.076352  3.974717  3.875617  3.778988
##  [71]  3.684767  3.592896  3.503315  3.415968  3.330798  3.247752  3.166777
##  [78]  3.087820  3.010832  2.935764  2.862567  2.791195  2.721603  2.653746
##  [85]  2.587581  2.523065  2.460158  2.398819  2.339010  2.280692  2.223828
##  [92]  2.168382  2.114318  2.061602  2.010201  1.960081  1.911210  1.863558
##  [99]  1.817095  1.771790  1.727614
## 
## 
## [[37]]
## [[37]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[37]]$y
##   [1]  0.000000  4.380762  7.712845 10.223383 12.090863 13.455524 14.427546
##   [8] 15.093474 15.521278 15.764329 15.864520 15.854722 15.760719 15.602721
##  [15] 15.396560 15.154628 14.886613 14.600078 14.300918 13.993720 13.682039
##  [22] 13.368625 13.055593 12.744561 12.436758 12.133109 11.834297 11.540823
##  [29] 11.253037 10.971181 10.695403 10.425787 10.162363  9.905118  9.654012
##  [36]  9.408977  9.169931  8.936777  8.709408  8.487712  8.271570  8.060864
##  [43]  7.855470  7.655267  7.460133  7.269946  7.084587  6.903938  6.727882
##  [50]  6.556306  6.389097  6.226146  6.067347  5.912594  5.761785  5.614820
##  [57]  5.471602  5.332035  5.196027  5.063488  4.934329  4.808464  4.685808
##  [64]  4.566282  4.449804  4.336296  4.225684  4.117894  4.012853  3.910491
##  [71]  3.810740  3.713534  3.618807  3.526497  3.436541  3.348880  3.263455
##  [78]  3.180209  3.099087  3.020033  2.942997  2.867925  2.794769  2.723478
##  [85]  2.654006  2.586306  2.520333  2.456043  2.393393  2.332341  2.272846
##  [92]  2.214869  2.158371  2.103314  2.049662  1.997378  1.946428  1.896777
##  [99]  1.848393  1.801243  1.755296
## 
## 
## [[38]]
## [[38]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[38]]$y
##   [1]  0.000000  4.029099  7.141078  9.524363 11.329117 12.675001 13.657357
##   [8] 14.352140 14.819851 15.108671 15.256960 15.295248 15.247826 15.134011
##  [15] 14.969153 14.765447 14.532566 14.278181 14.008360 13.727902 13.440589
##  [22] 13.149396 12.856658 12.564195 12.273422 11.985430 11.701053 11.420923
##  [29] 11.145507 10.875146 10.610080 10.350467 10.096404  9.847937  9.605076
##  [36]  9.367798  9.136057  8.909792  8.688924  8.473367  8.263027  8.057805
##  [43]  7.857596  7.662296  7.471796  7.285991  7.104773  6.928036  6.755673
##  [50]  6.587582  6.423660  6.263805  6.107921  5.955908  5.807674  5.663124
##  [57]  5.522169  5.384719  5.250689  5.119992  4.992548  4.868274  4.747093
##  [64]  4.628928  4.513704  4.401347  4.291787  4.184954  4.080780  3.979199
##  [71]  3.880146  3.783559  3.689376  3.597538  3.507985  3.420662  3.335512
##  [78]  3.252482  3.171519  3.092571  3.015589  2.940522  2.867325  2.795949
##  [85]  2.726350  2.658484  2.592307  2.527777  2.464854  2.403497  2.343667
##  [92]  2.285326  2.228438  2.172966  2.118875  2.066130  2.014699  1.964547
##  [99]  1.915644  1.867959  1.821460
## 
## 
## [[39]]
## [[39]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[39]]$y
##   [1]  0.000000  4.443035  7.772269 10.242844 12.051975 13.352094 14.261000
##   [8] 14.869721 15.248584 15.451915 15.521663 15.490214 15.382559 15.217972
##  [15] 15.011308 14.774008 14.514869 14.240654 13.956545 13.666508 13.373568
##  [22] 13.080024 12.787615 12.497646 12.211090 11.928662 11.650881 11.378110
##  [29] 11.110599 10.848507 10.591926 10.340892 10.095407  9.855438  9.620933
##  [36]  9.391822  9.168024  8.949448  8.735997  8.527570  8.324065  8.125377
##  [43]  7.931400  7.742031  7.557165  7.376699  7.200531  7.028562  6.860694
##  [50]  6.696830  6.536875  6.380739  6.228329  6.079558  5.934339  5.792588
##  [57]  5.654222  5.519160  5.387324  5.258637  5.133024  5.010411  4.890726
##  [64]  4.773901  4.659865  4.548554  4.439902  4.333845  4.230321  4.129270
##  [71]  4.030633  3.934352  3.840371  3.748635  3.659091  3.571685  3.486367
##  [78]  3.403087  3.321797  3.242448  3.164995  3.089391  3.015594  2.943560
##  [85]  2.873246  2.804612  2.737618  2.672223  2.608391  2.546084  2.485265
##  [92]  2.425898  2.367950  2.311386  2.256174  2.202280  2.149673  2.098323
##  [99]  2.048200  1.999274  1.951517
## 
## 
## [[40]]
## [[40]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[40]]$y
##   [1]  0.000000  4.357752  7.642091 10.093745 11.899974 13.206441 14.126432
##   [8] 14.748047 15.139783 15.354886 15.434737 15.411482 15.310085 15.149920
##  [15] 14.946014 14.710013 14.450928 14.175730 13.889795 13.597263 13.301312
##  [22] 13.004375 12.708303 12.414496 12.124005 11.837609 11.555879 11.279221
##  [29] 11.007917 10.742151 10.482034 10.227618  9.978913  9.735893  9.498511
##  [36]  9.266697  9.040371  8.819440  8.603805  8.393361  8.188003  7.987621
##  [43]  7.792105  7.601345  7.415232  7.233658  7.056517  6.883702  6.715112
##  [50]  6.550644  6.390199  6.233679  6.080991  5.932040  5.786736  5.644989
##  [57]  5.506714  5.371824  5.240238  5.111875  4.986656  4.864504  4.745343
##  [64]  4.629102  4.515708  4.405091  4.297184  4.191920  4.089235  3.989065
##  [71]  3.891349  3.796026  3.703038  3.612328  3.523841  3.437520  3.353315
##  [78]  3.271172  3.191041  3.112873  3.036620  2.962235  2.889671  2.818886
##  [85]  2.749834  2.682474  2.616764  2.552664  2.490133  2.429135  2.369631
##  [92]  2.311584  2.254959  2.199722  2.145837  2.093273  2.041996  1.991975
##  [99]  1.943179  1.895579  1.849145
## 
## 
## [[41]]
## [[41]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[41]]$y
##   [1]  0.000000  3.924752  6.983182  9.348826 11.160818 12.530705 13.547907
##   [8] 14.284099 14.796723 15.131811 15.326240 15.409547 15.405384 15.332684
##  [15] 15.206593 15.039225 14.840259 14.617423 14.376885 14.123555 13.861341
##  [22] 13.593347 13.322029 13.049330 12.776778 12.505568 12.236634 11.970695
##  [29] 11.708303 11.449873 11.195713 10.946046 10.701026 10.460752 10.225280
##  [36]  9.994635  9.768811  9.547785  9.331515  9.119947  8.913019  8.710660
##  [43]  8.512794  8.319341  8.130219  7.945345  7.764632  7.587996  7.415351
##  [50]  7.246613  7.081697  6.920521  6.763001  6.609058  6.458612  6.311585
##  [57]  6.167900  6.027483  5.890259  5.756157  5.625107  5.497038  5.371884
##  [64]  5.249578  5.130056  5.013255  4.899113  4.787569  4.678564  4.572041
##  [71]  4.467943  4.366215  4.266803  4.169655  4.074718  3.981943  3.891280
##  [78]  3.802681  3.716099  3.631489  3.548806  3.468004  3.389043  3.311879
##  [85]  3.236472  3.162783  3.090771  3.020398  2.951628  2.884424  2.818749
##  [92]  2.754570  2.691853  2.630563  2.570669  2.512138  2.454941  2.399045
##  [99]  2.344422  2.291043  2.238879
## 
## 
## [[42]]
## [[42]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[42]]$y
##   [1]  0.000000  4.232201  7.463987  9.909957 11.739157 13.084732 14.051547
##   [8] 14.722176 15.161635 15.421100 15.540846 15.552560 15.481158 15.346228
##  [15] 15.163155 14.944017 14.698286 14.433380 14.155103 13.867985 13.575555
##  [22] 13.280552 12.985097 12.690822 12.398974 12.110503 11.826118 11.546346
##  [29] 11.271565 11.002043 10.737957 10.479415 10.226473  9.979143  9.737406
##  [36]  9.501219  9.270520  9.045233  8.825271  8.610540  8.400941  8.196371
##  [43]  7.996724  7.801895  7.611777  7.426264  7.245249  7.068630  6.896302
##  [50]  6.728164  6.564117  6.404063  6.247907  6.095554  5.946912  5.801893
##  [57]  5.660408  5.522372  5.387701  5.256312  5.128128  5.003068  4.881058
##  [64]  4.762023  4.645891  4.532590  4.422052  4.314210  4.208998  4.106352
##  [71]  4.006209  3.908507  3.813189  3.720195  3.629469  3.540956  3.454601
##  [78]  3.370352  3.288158  3.207968  3.129734  3.053407  2.978942  2.906294
##  [85]  2.835416  2.766268  2.698805  2.632988  2.568777  2.506131  2.445012
##  [92]  2.385385  2.327211  2.270456  2.215086  2.161065  2.108362  2.056945
##  [99]  2.006781  1.957841  1.910094
## 
## 
## [[43]]
## [[43]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[43]]$y
##   [1]  0.000000  4.316658  7.651561 10.208745 12.150217 13.604608 14.674041
##   [8] 15.439581 15.965576 16.303092 16.492650 16.566397 16.549825 16.463149
##  [15] 16.322385 16.140222 15.926703 15.689772 15.435706 15.169458 14.894931
##  [22] 14.615197 14.332663 14.049213 13.766317 13.485111 13.206473 12.931074
##  [29] 12.659418 12.391882 12.128742 11.870190 11.616356 11.367320 11.123122
##  [36] 10.883774 10.649260 10.419549 10.194594  9.974338  9.758715  9.547654
##  [43]  9.341079  9.138910  8.941066  8.747467  8.558027  8.372666  8.191299
##  [50]  8.013845  7.840223  7.670352  7.504155  7.341552  7.182467  7.026825
##  [57]  6.874554  6.725579  6.579831  6.437239  6.297737  6.161257  6.027733
##  [64]  5.897103  5.769303  5.644272  5.521951  5.402280  5.285203  5.170663
##  [71]  5.058604  4.948975  4.841721  4.736791  4.634136  4.533705  4.435451
##  [78]  4.339326  4.245284  4.153280  4.063270  3.975211  3.889060  3.804777
##  [85]  3.722319  3.641649  3.562727  3.485516  3.409978  3.336077  3.263777
##  [92]  3.193045  3.123845  3.056145  2.989912  2.925115  2.861722  2.799703
##  [99]  2.739027  2.679667  2.621593
## 
## 
## [[44]]
## [[44]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[44]]$y
##   [1]  0.000000  4.494030  7.944662 10.570272 12.544080 14.003468 15.057388
##   [8] 15.792226 16.276465 16.564386 16.698995 16.714359 16.637447 16.489603
##  [15] 16.287704 16.045084 15.772272 15.477565 15.167497 14.847199 14.520693
##  [22] 14.191123 13.860934 13.532021 13.205841 12.883508 12.565860 12.253522
##  [29] 11.946949 11.646462 11.352276 11.064524 10.783276 10.508549 10.240325
##  [36]  9.978552  9.723157  9.474051  9.231130  8.994281  8.763385  8.538318
##  [43]  8.318953  8.105162  7.896816  7.693786  7.495946  7.303168  7.115329
##  [50]  6.932305  6.753977  6.580226  6.410938  6.245999  6.085298  5.928728
##  [57]  5.776184  5.627561  5.482762  5.341686  5.204239  5.070328  4.939861
##  [64]  4.812752  4.688912  4.568259  4.450709  4.336185  4.224607  4.115900
##  [71]  4.009990  3.906806  3.806276  3.708333  3.612911  3.519943  3.429368
##  [78]  3.341124  3.255150  3.171389  3.089783  3.010276  2.932816  2.857349
##  [85]  2.783823  2.712190  2.642400  2.574406  2.508161  2.443621  2.380742
##  [92]  2.319481  2.259796  2.201647  2.144994  2.089799  2.036024  1.983633
##  [99]  1.932591  1.882861  1.834411
## 
## 
## [[45]]
## [[45]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[45]]$y
##   [1]  0.000000  3.999934  7.067121  9.398507 11.149909 12.444589 13.380047
##   [8] 14.033397 14.465614 14.724906 14.849376 14.869130 14.807948 14.684601
##  [15] 14.513900 14.307522 14.074663 13.822560 13.556897 13.282132 13.001753
##  [22] 12.718482 12.434434 12.151246 11.870179 11.592193 11.318014 11.048184
##  [29] 10.783098 10.523036 10.268190 10.018680  9.774572  9.535889  9.302620
##  [36]  9.074730  8.852163  8.634851  8.422710  8.215653  8.013585  7.816406
##  [43]  7.624015  7.436309  7.253185  7.074539  6.900268  6.730270  6.564444
##  [50]  6.402692  6.244916  6.091020  5.940910  5.794495  5.651684  5.512391
##  [57]  5.376527  5.244011  5.114759  4.988692  4.865731  4.745800  4.628824
##  [64]  4.514731  4.403450  4.294912  4.189049  4.085795  3.985086  3.886859
##  [71]  3.791053  3.697608  3.606467  3.517572  3.430869  3.346302  3.263820
##  [78]  3.183371  3.104905  3.028373  2.953727  2.880922  2.809911  2.740650
##  [85]  2.673096  2.607208  2.542943  2.480263  2.419127  2.359499  2.301340
##  [92]  2.244615  2.189288  2.135325  2.082692  2.031356  1.981286  1.932449
##  [99]  1.884817  1.838358  1.793045
## 
## 
## [[46]]
## [[46]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[46]]$y
##   [1]  0.000000  4.036784  7.117696  9.448497 11.191099 12.472897 13.394122
##   [8] 14.033618 14.453400 14.702234 14.818457 14.832203 14.767144 14.641871
##  [15] 14.470973 14.265893 14.035596 13.787098 13.525884 13.256231 12.981470
##  [22] 12.704185 12.426374 12.149575 11.874966 11.603439 11.335664 11.072137
##  [29] 10.813218 10.559159 10.310129 10.066232  9.827523  9.594017  9.365699
##  [36]  9.142531  8.924457  8.711409  8.503309  8.300071  8.101605  7.907817
##  [43]  7.718612  7.533891  7.353559  7.177517  7.005669  6.837919  6.674174
##  [50]  6.514340  6.358326  6.206042  6.057400  5.912315  5.770702  5.632478
##  [57]  5.497563  5.365879  5.237348  5.111894  4.989445  4.869928  4.753274
##  [64]  4.639414  4.528281  4.419810  4.313937  4.210600  4.109738  4.011292
##  [71]  3.915204  3.821418  3.729879  3.640532  3.553326  3.468208  3.385129
##  [78]  3.304041  3.224895  3.147644  3.072245  2.998651  2.926820  2.856710
##  [85]  2.788280  2.721488  2.656297  2.592667  2.530561  2.469943  2.410777
##  [92]  2.353029  2.296663  2.241648  2.187951  2.135540  2.084384  2.034454
##  [99]  1.985720  1.938154  1.891726
## 
## 
## [[47]]
## [[47]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[47]]$y
##   [1]  0.000000  4.586780  8.033807 10.600235 12.486801 13.848979 14.807177
##   [8] 15.454635 15.863540 16.089775 16.176584 16.157428 16.058181 15.898846
##  [15] 15.694874 15.458193 15.198000 14.921379 14.633778 14.339375 14.041369
##  [22] 13.742200 13.443720 13.147327 12.854067 12.564718 12.279845 11.999854
##  [29] 11.725026 11.455546 11.191528 10.933028 10.680059 10.432602 10.190616
##  [36]  9.954038  9.722795  9.496800  9.275964  9.060189  8.849377  8.643426
##  [43]  8.442233  8.245697  8.053716  7.866189  7.683016  7.504099  7.329341
##  [50]  7.158647  6.991923  6.829080  6.670026  6.514675  6.362940  6.214739
##  [57]  6.069988  5.928608  5.790520  5.655648  5.523917  5.395254  5.269588
##  [64]  5.146849  5.026968  4.909880  4.795518  4.683821  4.574725  4.468170
##  [71]  4.364097  4.262447  4.163166  4.066197  3.971487  3.878982  3.788632
##  [78]  3.700387  3.614197  3.530015  3.447793  3.367487  3.289051  3.212442
##  [85]  3.137617  3.064535  2.993156  2.923439  2.855346  2.788839  2.723880
##  [92]  2.660435  2.598468  2.537944  2.478830  2.421093  2.364701  2.309622
##  [99]  2.255826  2.203283  2.151963
## 
## 
## [[48]]
## [[48]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[48]]$y
##   [1]  0.000000  4.757118  8.302703 10.918282 12.820566 14.176356 15.114027
##   [8] 15.732362 16.107362 16.297488 16.347691 16.292528 16.158547 15.966135
##  [15] 15.730935 15.464936 15.177319 14.875099 14.563631 14.246986 13.928251
##  [22] 13.609759 13.293257 12.980049 12.671092 12.367085 12.068521 11.775740
##  [29] 11.488967 11.208335 10.933908 10.665700 10.403686 10.147814  9.898006
##  [36]  9.654172  9.416209  9.184008  8.957453  8.736425  8.520803  8.310466
##  [43]  8.105294  7.905165  7.709961  7.519564  7.333859  7.152732  6.976074
##  [50]  6.803773  6.635725  6.471825  6.311971  6.156064  6.004006  5.855704
##  [57]  5.711064  5.569996  5.432412  5.298226  5.167355  5.039716  4.915230
##  [64]  4.793818  4.675405  4.559918  4.447283  4.337430  4.230290  4.125797
##  [71]  4.023885  3.924490  3.827551  3.733006  3.640796  3.550864  3.463154
##  [78]  3.377610  3.294179  3.212809  3.133449  3.056049  2.980561  2.906938
##  [85]  2.835133  2.765102  2.696800  2.630186  2.565218  2.501854  2.440055
##  [92]  2.379783  2.320999  2.263668  2.207753  2.153218  2.100031  2.048158
##  [99]  1.997566  1.948224  1.900101
## 
## 
## [[49]]
## [[49]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[49]]$y
##   [1]  0.000000  4.117735  7.278760  9.686398 11.501137 12.849647 13.831897
##   [8] 14.526775 14.996527 15.290259 15.446704 15.496408 15.463451 15.366809
##  [15] 15.221430 15.039083 14.829026 14.598537 14.353329 14.097885 13.835711
##  [22] 13.569546 13.301525 13.033305 12.766165 12.501090 12.238830 11.979950
##  [29] 11.724874 11.473908 11.227273 10.985118 10.747537 10.514582 10.286272
##  [36] 10.062601  9.843541  9.629052  9.419079  9.213561  9.012430  8.815612
##  [43]  8.623031  8.434609  8.250266  8.069923  7.893497  7.720910  7.552082
##  [50]  7.386934  7.225388  7.067367  6.912797  6.761603  6.613712  6.469053
##  [57]  6.327556  6.189152  6.053774  5.921356  5.791834  5.665144  5.541225
##  [64]  5.420016  5.301458  5.185493  5.072064  4.961117  4.852596  4.746449
##  [71]  4.642624  4.541070  4.441737  4.344577  4.249542  4.156586  4.065664
##  [78]  3.976730  3.889741  3.804656  3.721431  3.640027  3.560404  3.482523
##  [85]  3.406345  3.331833  3.258951  3.187664  3.117936  3.049733  2.983022
##  [92]  2.917770  2.853946  2.791517  2.730455  2.670728  2.612307  2.555165
##  [99]  2.499272  2.444602  2.391128
## 
## 
## [[50]]
## [[50]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[50]]$y
##   [1]  0.000000  3.557213  6.308645  8.418979 10.019637 11.215488 12.090199
##   [8] 12.710518 13.129683 13.390147 13.525758 13.563490 13.524836 13.426909
##  [15] 13.283329 13.104929 12.900316 12.676323 12.438367 12.190733 11.936808
##  [22] 11.679258 11.420178 11.161207 10.903619 10.648400 10.396305 10.147906
##  [29]  9.903632  9.663793  9.428609  9.198230  8.972746  8.752205  8.536617
##  [36]  8.325966  8.120216  7.919313  7.723191  7.531775  7.344981  7.162724
##  [43]  6.984913  6.811454  6.642254  6.477217  6.316250  6.159258  6.006148
##  [50]  5.856829  5.711208  5.569199  5.430712  5.295663  5.163967  5.035541
##  [57]  4.910307  4.788184  4.669097  4.552969  4.439729  4.329304  4.221625
##  [64]  4.116623  4.014232  3.914388  3.817026  3.722086  3.629507  3.539231
##  [71]  3.451200  3.365359  3.281652  3.200028  3.120433  3.042819  2.967135
##  [78]  2.893333  2.821367  2.751191  2.682760  2.616032  2.550963  2.487513
##  [85]  2.425641  2.365307  2.306475  2.249106  2.193163  2.138613  2.085419
##  [92]  2.033548  1.982967  1.933645  1.885549  1.838650  1.792917  1.748321
##  [99]  1.704835  1.662431  1.621081
## 
## 
## [[51]]
## [[51]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[51]]$y
##   [1]  0.000000  4.096079  7.249990  9.656727 11.471430 12.817515 13.793149
##   [8] 14.476405 14.929361 15.201367 15.331643 15.351347 15.285221 15.152904
##  [15] 14.969971 14.748766 14.499063 14.228589 13.943446 13.648443 13.347360
##  [22] 13.043159 12.738154 12.434143 12.132514 11.834333 11.540404 11.251332
##  [29] 10.967558 10.689395 10.417058 10.150682  9.890340  9.636056  9.387818
##  [36]  9.145582  8.909286  8.678846  8.454169  8.235152  8.021683  7.813649
##  [43]  7.610930  7.413407  7.220961  7.033470  6.850816  6.672880  6.499545
##  [50]  6.330697  6.166222  6.006011  5.849953  5.697945  5.549881  5.405660
##  [57]  5.265185  5.128357  4.995082  4.865270  4.738830  4.615674  4.495719
##  [64]  4.378880  4.265078  4.154232  4.046267  3.941108  3.838682  3.738917
##  [71]  3.641745  3.547098  3.454911  3.365120  3.277663  3.192478  3.109507
##  [78]  3.028693  2.949979  2.873310  2.798634  2.725899  2.655054  2.586051
##  [85]  2.518841  2.453377  2.389615  2.327510  2.267020  2.208101  2.150713
##  [92]  2.094817  2.040374  1.987346  1.935696  1.885388  1.836388  1.788661
##  [99]  1.742174  1.696896  1.652795
## 
## 
## [[52]]
## [[52]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[52]]$y
##   [1]  0.000000  3.891155  6.898746  9.206180 10.959128 12.273283 13.240523
##   [8] 13.933804 14.411053 14.718256 14.891912 14.960986 14.948457 14.872548
##  [15] 14.747707 14.585380 14.394633 14.182637 13.955061 13.716382 13.470128
##  [22] 13.219076 12.965405 12.710821 12.456654 12.203937 11.953466 11.705848
##  [29] 11.461545 11.220901 10.984165 10.751516 10.523073 10.298912 10.079071
##  [36]  9.863561  9.652372  9.445477  9.242836  9.044398  8.850106  8.659898
##  [43]  8.473705  8.291458  8.113086  7.938514  7.767670  7.600480  7.436870
##  [50]  7.276768  7.120101  6.966798  6.816789  6.670004  6.526375  6.385835
##  [57]  6.248319  6.113762  5.982100  5.853273  5.727219  5.603878  5.483194
##  [64]  5.365107  5.249563  5.136508  5.025886  4.917647  4.811739  4.708112
##  [71]  4.606716  4.507503  4.410428  4.315443  4.222503  4.131566  4.042586
##  [78]  3.955523  3.870335  3.786981  3.705423  3.625621  3.547538  3.471136
##  [85]  3.396380  3.323234  3.251663  3.181634  3.113112  3.046067  2.980465
##  [92]  2.916276  2.853470  2.792016  2.731886  2.673050  2.615482  2.559154
##  [99]  2.504038  2.450110  2.397343
## 
## 
## [[53]]
## [[53]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[53]]$y
##   [1]  0.000000  4.018107  7.109334  9.467257 11.245464 12.565808 13.524952
##   [8] 14.199568 14.650458 14.925835 15.063913 15.094978 15.043020 14.927034
##  [15] 14.762051 14.559957 14.330144 14.080024 13.815438 13.540983 13.260268
##  [22] 12.976118 12.690740 12.405848 12.122767 11.842514 11.565862 11.293391
##  [29] 11.025530 10.762589 10.504780 10.252244 10.005063  9.763274  9.526877
##  [36]  9.295846  9.070134  8.849677  8.634399  8.424217  8.219038  8.018767
##  [43]  7.823306  7.632552  7.446405  7.264762  7.087522  6.914583  6.745847
##  [50]  6.581214  6.420588  6.263874  6.110977  5.961807  5.816274  5.674290
##  [57]  5.535769  5.400628  5.268784  5.140157  5.014669  4.892244  4.772807
##  [64]  4.656286  4.542608  4.431706  4.323511  4.217957  4.114980  4.014517
##  [71]  3.916506  3.820888  3.727605  3.636599  3.547815  3.461198  3.376696
##  [78]  3.294257  3.213830  3.135367  3.058820  2.984142  2.911286  2.840210
##  [85]  2.770868  2.703220  2.637223  2.572838  2.510024  2.448744  2.388960
##  [92]  2.330635  2.273735  2.218224  2.164068  2.111234  2.059690  2.009404
##  [99]  1.960346  1.912486  1.865794
## 
## 
## [[54]]
## [[54]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[54]]$y
##   [1]  0.000000  4.119424  7.241156  9.585482 11.324510 12.592690 13.495046
##   [8] 14.113608 14.512431 14.741531 14.839945 14.838139 14.759874 14.623678
##  [15] 14.443987 14.232040 13.996582 13.744403 13.480775 13.209776 12.934558
##  [22] 12.657547 12.380603 12.105146 11.832250 11.562722 11.297161 11.036003
##  [29] 10.779556 10.528035 10.281573 10.040250  9.804098  9.573114  9.347272
##  [36]  9.126521  8.910801  8.700035  8.494143  8.293035  8.096620  7.904804
##  [43]  7.717489  7.534581  7.355982  7.181597  7.011330  6.845088  6.682778
##  [50]  6.524309  6.369592  6.218540  6.071066  5.927086  5.786519  5.649284
##  [57]  5.515303  5.384498  5.256794  5.132119  5.010400  4.891567  4.775553
##  [64]  4.662289  4.551712  4.443758  4.338363  4.235468  4.135014  4.036942
##  [71]  3.941196  3.847721  3.756463  3.667369  3.580388  3.495470  3.412566
##  [78]  3.331629  3.252611  3.175467  3.100153  3.026625  2.954841  2.884760
##  [85]  2.816340  2.749544  2.684332  2.620666  2.558510  2.497829  2.438587
##  [92]  2.380749  2.324284  2.269158  2.215339  2.162797  2.111501  2.061421
##  [99]  2.012529  1.964797  1.918197
## 
## 
## [[55]]
## [[55]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[55]]$y
##   [1]  0.000000  4.352245  7.679843 10.203178 12.095656 13.493713 14.504685
##   [8] 15.213006 15.685087 15.973151 16.118261 16.152693 16.101808 15.985527
##  [15] 15.819491 15.615969 15.384584 15.132869 14.866719 14.590739 14.308517
##  [22] 14.022846 13.735888 13.449316 13.164413 12.882160 12.603296 12.328376
##  [29] 12.057808 11.791883 11.530805 11.274707 11.023666 10.777719 10.536869
##  [36] 10.301095 10.070354  9.844592  9.623742  9.407728  9.196470  8.989883
##  [43]  8.787880  8.590370  8.397264  8.208470  8.023900  7.843462  7.667068
##  [50]  7.494630  7.326062  7.161279  7.000197  6.842734  6.688810  6.538346
##  [57]  6.391264  6.247490  6.106948  5.969567  5.835276  5.704006  5.575687
##  [64]  5.450256  5.327645  5.207793  5.090637  4.976116  4.864171  4.754745
##  [71]  4.647780  4.543221  4.441015  4.341108  4.243448  4.147986  4.054671
##  [78]  3.963455  3.874291  3.787133  3.701936  3.618655  3.537248  3.457673
##  [85]  3.379887  3.303852  3.229527  3.156874  3.085855  3.016434  2.948575
##  [92]  2.882242  2.817402  2.754020  2.692065  2.631503  2.572303  2.514435
##  [99]  2.457869  2.402576  2.348526
## 
## 
## [[56]]
## [[56]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[56]]$y
##   [1]  0.000000  4.707886  8.292255 10.997081 13.013878 14.492968 15.552326
##   [8] 16.284533 16.762228 17.042394 17.169721 17.179244 17.098426 16.948775
##  [15] 16.747134 16.506678 16.237703 15.948249 15.644583 15.331582 15.013030
##  [22] 14.691858 14.370325 14.050164 13.732698 13.418925 13.109593 12.805252
##  [29] 12.506296 12.213002 11.925552 11.644056 11.368567 11.099096 10.835619
##  [36] 10.578089 10.326437 10.080582  9.840432  9.605885  9.376837  9.153178
##  [43]  8.934797  8.721582  8.513419  8.310198  8.111806  7.918134  7.729072
##  [50]  7.544514  7.364355  7.188492  7.016823  6.849249  6.685675  6.526005
##  [57]  6.370146  6.218008  6.069502  5.924542  5.783043  5.644924  5.510102
##  [64]  5.378501  5.250042  5.124651  5.002255  4.882782  4.766162  4.652327
##  [71]  4.541212  4.432750  4.326878  4.223535  4.122661  4.024195  3.928082
##  [78]  3.834263  3.742686  3.653296  3.566041  3.480870  3.397733  3.316581
##  [85]  3.237368  3.160047  3.084573  3.010901  2.938988  2.868794  2.800276
##  [92]  2.733394  2.668110  2.604385  2.542182  2.481464  2.422197  2.364345
##  [99]  2.307875  2.252754  2.198949
## 
## 
## [[57]]
## [[57]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[57]]$y
##   [1]  0.000000  3.703077  6.539096  8.693180 10.311297 11.508532 12.375612
##   [8] 12.984057 13.390235 13.638566 13.764045 13.794239 13.750850 13.650963
##  [15] 13.508016 13.332574 13.132936 12.915615 12.685716 12.447235 12.203292
##  [22] 11.956318 11.708202 11.460405 11.214051 10.969999 10.728901 10.491245
##  [29] 10.257389 10.027593  9.802036  9.580835  9.364061  9.151746  8.943894
##  [36]  8.740486  8.541484  8.346842  8.156499  7.970389  7.788442  7.610581
##  [43]  7.436729  7.266808  7.100736  6.938434  6.779821  6.624818  6.473346
##  [50]  6.325327  6.180685  6.039344  5.901231  5.766272  5.634397  5.505535
##  [57]  5.379619  5.256581  5.136356  5.018879  4.904089  4.791923  4.682323
##  [64]  4.575229  4.470584  4.368332  4.268419  4.170791  4.075396  3.982182
##  [71]  3.891101  3.802103  3.715140  3.630166  3.547136  3.466005  3.386729
##  [78]  3.309267  3.233576  3.159617  3.087349  3.016734  2.947735  2.880313
##  [85]  2.814434  2.750061  2.687161  2.625699  2.565643  2.506961  2.449621
##  [92]  2.393593  2.338846  2.285351  2.233080  2.182004  2.132096  2.083330
##  [99]  2.035680  1.989119  1.943623
## 
## 
## [[58]]
## [[58]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[58]]$y
##   [1]  0.000000  4.030978  7.117266  9.459521 11.216233 12.512578 13.447409
##   [8] 14.098778 14.528308 14.784637 14.906146 14.923112 14.859411 14.733860
##  [15] 14.561282 14.353344 14.119220 13.866118 13.599690 13.324361 13.043587
##  [22] 12.760062 12.475874 12.192638 11.911593 11.633682 11.359617 11.089927
##  [29] 10.824996 10.565096 10.310412 10.061058  9.817095  9.578543  9.345389
##  [36]  9.117596  8.895106  8.677848  8.465742  8.258697  8.056619  7.859409
##  [43]  7.666965  7.479186  7.295968  7.117209  6.942806  6.772658  6.606665
##  [50]  6.444728  6.286752  6.132641  5.982302  5.835643  5.692577  5.553015
##  [57]  5.416873  5.284066  5.154515  5.028138  4.904859  4.784602  4.667292
##  [64]  4.552859  4.441231  4.332339  4.226117  4.122500  4.021422  3.922823
##  [71]  3.826642  3.732818  3.641295  3.552016  3.464926  3.379971  3.297099
##  [78]  3.216259  3.137401  3.060476  2.985438  2.912239  2.840835  2.771182
##  [85]  2.703236  2.636957  2.572303  2.509233  2.447711  2.387696  2.329153
##  [92]  2.272046  2.216338  2.161997  2.108988  2.057279  2.006837  1.957632
##  [99]  1.909634  1.862812  1.817139
## 
## 
## [[59]]
## [[59]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[59]]$y
##   [1]  0.000000  3.711400  6.582247  8.785281 10.458113 11.710361 12.629333
##   [8] 13.284561 13.731415 14.013973 14.167324 14.219389 14.192377 14.103952
##  [15] 13.968150 13.796121 13.596714 13.376948 13.142380 12.897406 12.645497
##  [22] 12.389386 12.131223 11.872688 11.615091 11.359449 11.106540 10.856962
##  [29] 10.611159 10.369463 10.132112  9.899268  9.671038  9.447483  9.228629
##  [36]  9.014472  8.804988  8.600135  8.399860  8.204099  8.012781  7.825829
##  [43]  7.643163  7.464699  7.290355  7.120044  6.953682  6.791182  6.632460
##  [50]  6.477432  6.326015  6.178128  6.033691  5.892624  5.754850  5.620293
##  [57]  5.488880  5.360536  5.235192  5.112777  4.993223  4.876464  4.762434
##  [64]  4.651069  4.542309  4.436091  4.332357  4.231048  4.132108  4.035482
##  [71]  3.941115  3.848954  3.758949  3.671048  3.585203  3.501365  3.419488
##  [78]  3.339525  3.261432  3.185165  3.110682  3.037940  2.966899  2.897520
##  [85]  2.829763  2.763591  2.698966  2.635852  2.574214  2.514017  2.455228
##  [92]  2.397814  2.341742  2.286982  2.233502  2.181273  2.130265  2.080450
##  [99]  2.031799  1.984287  1.937885
## 
## 
## [[60]]
## [[60]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[60]]$y
##   [1]  0.000000  4.439810  7.838953 10.419137 12.355309 13.785510 14.818646
##   [8] 15.540630 16.019224 16.307864 16.448679 16.474873 16.412609 16.282489
##  [15] 16.100730 15.880087 15.630584 15.360090 15.074774 14.779463 14.477929
##  [22] 14.173107 13.867278 13.562203 13.259236 12.959409 12.663503 12.372100
##  [29] 12.085624 11.804379 11.528573 11.258338 10.993749 10.734835 10.481590
##  [36] 10.233980  9.991953  9.755439  9.524357  9.298617  9.078124  8.862777
##  [43]  8.652475  8.447111  8.246582  8.050781  7.859604  7.672947  7.490707
##  [50]  7.312784  7.139077  6.969488  6.803922  6.642284  6.484482  6.330427
##  [57]  6.180029  6.033202  5.889862  5.749927  5.613315  5.479948  5.349750
##  [64]  5.222644  5.098558  4.977420  4.859159  4.743709  4.631001  4.520971
##  [71]  4.413555  4.308692  4.206319  4.106379  4.008814  3.913566  3.820582
##  [78]  3.729807  3.641188  3.554675  3.470218  3.387767  3.307276  3.228696
##  [85]  3.151984  3.077094  3.003984  2.932611  2.862933  2.794911  2.728505
##  [92]  2.663677  2.600390  2.538606  2.478290  2.419407  2.361923  2.305804
##  [99]  2.251020  2.197536  2.145324
## 
## 
## [[61]]
## [[61]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[61]]$y
##   [1]  0.000000  4.380904  7.715807 10.231923 12.107592 13.482790 14.467362
##   [8] 15.147502 15.590833 15.850397 15.967792 15.975627 15.899457 15.759299
##  [15] 15.570823 15.346286 15.095269 14.825247 14.542051 14.250215 13.953258
##  [22] 13.653904 13.354253 13.055916 12.760120 12.467793 12.179628 11.896135
##  [29] 11.617682 11.344522 11.076825 10.814690 10.558168 10.307266 10.061961
##  [36]  9.822209  9.587944  9.359091  9.135561  8.917262  8.704093  8.495954
##  [43]  8.292739  8.094343  7.900661  7.711588  7.527020  7.346854  7.170988
##  [50]  6.999322  6.831758  6.668199  6.508552  6.352723  6.200622  6.052160
##  [57]  5.907251  5.765811  5.627756  5.493005  5.361481  5.233105  5.107802
##  [64]  4.985500  4.866125  4.749609  4.635883  4.524879  4.416534  4.310782
##  [71]  4.207563  4.106815  4.008479  3.912498  3.818816  3.727376  3.638126
##  [78]  3.551013  3.465985  3.382994  3.301990  3.222925  3.145754  3.070430
##  [85]  2.996910  2.925151  2.855110  2.786745  2.720018  2.654889  2.591319
##  [92]  2.529271  2.468708  2.409596  2.351900  2.295585  2.240618  2.186967
##  [99]  2.134602  2.083490  2.033601
## 
## 
## [[62]]
## [[62]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[62]]$y
##   [1]  0.000000  4.448875  7.812001 10.332333 12.198909 13.558803 14.526431
##   [8] 15.190794 15.621115 15.871224 15.982970 15.988884 15.914241 15.778671
##  [15] 15.597414 15.382290 15.142461 14.885022 14.615458 14.338001 14.055913
##  [22] 13.771698 13.487270 13.204088 12.923256 12.645603 12.371743 12.102123
##  [29] 11.837066 11.576792 11.321446 11.071113 10.825834 10.585615 10.350435
##  [36] 10.120252  9.895013  9.674649  9.459086  9.248242  9.042034  8.840373
##  [43]  8.643170  8.450336  8.261781  8.077414  7.897147  7.720892  7.548562
##  [50]  7.380072  7.215337  7.054275  6.896806  6.742849  6.592326  6.445163
##  [57]  6.301283  6.160615  6.023086  5.888627  5.757169  5.628645  5.502990
##  [64]  5.380140  5.260033  5.142606  5.027801  4.915559  4.805823  4.698536
##  [71]  4.593644  4.491094  4.390834  4.292811  4.196977  4.103282  4.011679
##  [78]  3.922121  3.834562  3.748958  3.665265  3.583440  3.503442  3.425230
##  [85]  3.348764  3.274005  3.200915  3.129457  3.059594  2.991290  2.924512
##  [92]  2.859224  2.795394  2.732988  2.671976  2.612326  2.554007  2.496991
##  [99]  2.441247  2.386748  2.333465
## 
## 
## [[63]]
## [[63]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[63]]$y
##   [1]  0.000000  3.851673  6.834279  9.126347 10.870087 12.178770 13.142609
##   [8] 13.833446 14.308483 14.613264 14.784038 14.849651 14.833053 14.752494
##  [15] 14.622482 14.454542 14.257825 14.039589 13.805587 13.560372 13.307540
##  [22] 13.049928 12.789767 12.528806 12.268414 12.009650 11.753336 11.500097
##  [29] 11.250409 11.004626 10.763004 10.525725 10.292911 10.064637  9.840939
##  [36]  9.621826  9.407283  9.197278  8.991766  8.790690  8.593986  8.401585
##  [43]  8.213412  8.029391  7.849443  7.673487  7.501445  7.333234  7.168775
##  [50]  7.007988  6.850795  6.697117  6.546878  6.400004  6.256419  6.116051
##  [57]  5.978830  5.844684  5.713546  5.585349  5.460027  5.337516  5.217753
##  [64]  5.100677  4.986227  4.874344  4.764972  4.658053  4.553534  4.451359
##  [71]  4.351477  4.253837  4.158386  4.065078  3.973863  3.884695  3.797528
##  [78]  3.712317  3.629017  3.547587  3.467984  3.390167  3.314096  3.239733
##  [85]  3.167037  3.095973  3.026504  2.958593  2.892206  2.827309  2.763868
##  [92]  2.701850  2.641224  2.581959  2.524023  2.467387  2.412022  2.357900
##  [99]  2.304992  2.253271  2.202710
## 
## 
## [[64]]
## [[64]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[64]]$y
##   [1]  0.000000  4.334097  7.604651 10.052110 11.862958 13.181817 14.120837
##   [8] 14.766977 15.187659 15.435147 15.549946 15.563446 15.499960 15.378316
##  [15] 15.213086 15.015541 14.794395 14.556374 14.306668 14.049271 13.787255
##  [22] 13.522971 13.258218 12.994363 12.732440 12.473225 12.217294 11.965070
##  [29] 11.716856 11.472862 11.233229 10.998043 10.767348 10.541157 10.319456
##  [36] 10.102216  9.889393  9.680932  9.476770  9.276841  9.081073  8.889393
##  [43]  8.701724  8.517990  8.338116  8.162024  7.989639  7.820884  7.655687
##  [50]  7.493973  7.335671  7.180709  7.029019  6.880530  6.735177  6.592893
##  [57]  6.453613  6.317276  6.183818  6.053179  5.925300  5.800122  5.677588
##  [64]  5.557642  5.440231  5.325300  5.212797  5.102670  4.994870  4.889348
##  [71]  4.786054  4.684943  4.585968  4.489084  4.394247  4.301413  4.210541
##  [78]  4.121588  4.034514  3.949280  3.865847  3.784176  3.704231  3.625974
##  [85]  3.549371  3.474387  3.400986  3.329136  3.258804  3.189958  3.122566
##  [92]  3.056598  2.992024  2.928814  2.866939  2.806371  2.747083  2.689048
##  [99]  2.632238  2.576629  2.522195
## 
## 
## [[65]]
## [[65]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[65]]$y
##   [1]  0.000000  4.268804  7.493144  9.906204 11.689588 12.984717 13.901709
##   [8] 14.526298 14.925222 15.150422 15.242314 15.232335 15.144929 14.999091
##  [15] 14.809578 14.587838 14.342751 14.081191 13.808471 13.528695 13.245016
##  [22] 12.959856 12.675064 12.392044 12.111855 11.835286 11.562921 11.295180
##  [29] 11.032359 10.774657 10.522199 10.275052 10.033239  9.796751  9.565551
##  [36]  9.339586  9.118784  8.903069  8.692351  8.486538  8.285535  8.089242
##  [43]  7.897562  7.710393  7.527637  7.349194  7.174967  7.004859  6.838775
##  [50]  6.676623  6.518310  6.363747  6.212846  6.065520  5.921686  5.781261
##  [57]  5.644166  5.510320  5.379647  5.252073  5.127524  5.005928  4.887215
##  [64]  4.771318  4.658168  4.547702  4.439855  4.334566  4.231774  4.131419
##  [71]  4.033444  3.937792  3.844409  3.753240  3.664234  3.577338  3.492503
##  [78]  3.409679  3.328820  3.249878  3.172809  3.097567  3.024109  2.952393
##  [85]  2.882378  2.814024  2.747290  2.682139  2.618533  2.556436  2.495811
##  [92]  2.436624  2.378840  2.322427  2.267351  2.213582  2.161087  2.109838
##  [99]  2.059804  2.010957  1.963267
## 
## 
## [[66]]
## [[66]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[66]]$y
##   [1]  0.000000  4.495723  7.918146 10.500917 12.427300 13.841024 14.854801
##   [8] 15.557011 16.016943 16.288911 16.415479 16.429991 16.358565 16.221645
##  [15] 16.035224 15.811809 15.561168 15.290924 15.007015 14.714062 14.415651
##  [22] 14.114556 13.812917 13.512378 13.214190 12.919303 12.628426 12.342084
##  [29] 12.060656 11.784407 11.513514 11.248087 10.988182 10.733814 10.484967
##  [36] 10.241601 10.003657  9.771065  9.543741  9.321596  9.104537  8.892465
##  [43]  8.685280  8.482881  8.285166  8.092033  7.903383  7.719115  7.539132
##  [50]  7.363335  7.191629  7.023922  6.860121  6.700136  6.543880  6.391265
##  [57]  6.242208  6.096625  5.954437  5.815564  5.679930  5.547458  5.418076
##  [64]  5.291710  5.168292  5.047752  4.930023  4.815040  4.702738  4.593056
##  [71]  4.485932  4.381306  4.279120  4.179317  4.081843  3.986641  3.893660
##  [78]  3.802848  3.714154  3.627528  3.542922  3.460290  3.379585  3.300763
##  [85]  3.223779  3.148590  3.075155  3.003433  2.933383  2.864967  2.798147
##  [92]  2.732886  2.669146  2.606893  2.546092  2.486709  2.428711  2.372066
##  [99]  2.316742  2.262708  2.209935
## 
## 
## [[67]]
## [[67]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[67]]$y
##   [1]  0.000000  4.252872  7.461626  9.860095 11.630235 12.913612 13.820364
##   [8] 14.436166 14.827650 15.046629 15.133384 15.119217 15.028448 14.879956
##  [15] 14.688396 14.465124 14.218942 13.956655 13.683519 13.403587 13.119972
##  [22] 12.835059 12.550668 12.268178 11.988629 11.712791 11.441235 11.174367
##  [29] 10.912475 10.655748 10.404305 10.158206  9.917471  9.682084  9.452007
##  [36]  9.227181  9.007535  8.792987  8.583447  8.378821  8.179012  7.983920
##  [43]  7.793444  7.607483  7.425937  7.248706  7.075691  6.906794  6.741921
##  [50]  6.580978  6.423871  6.270511  6.120809  5.974679  5.832035  5.692796
##  [57]  5.556880  5.424208  5.294703  5.168290  5.044894  4.924444  4.806870
##  [64]  4.692102  4.580075  4.470722  4.363980  4.259786  4.158081  4.058803
##  [71]  3.961896  3.867302  3.774967  3.684837  3.596858  3.510980  3.427152
##  [78]  3.345326  3.265453  3.187488  3.111384  3.037097  2.964584  2.893802
##  [85]  2.824710  2.757267  2.691435  2.627175  2.564449  2.503220  2.443454
##  [92]  2.385114  2.328168  2.272581  2.218321  2.165356  2.113657  2.063191
##  [99]  2.013931  1.965846  1.918910
## 
## 
## [[68]]
## [[68]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[68]]$y
##   [1]  0.000000  3.825583  6.772739  9.023499 10.722617 11.985184 12.902706
##   [8] 13.547918 13.978630 14.240779 14.370853 14.397833 14.344722 14.229771
##  [15] 14.067453 13.869237 13.644198 13.399516 13.140858 12.872692 12.598530
##  [22] 12.321129 12.042642 11.764744 11.488733 11.215604 10.946115 10.680834
##  [29] 10.420181 10.164457  9.913871  9.668558  9.428595  9.194016  8.964819
##  [36]  8.740973  8.522428  8.309119  8.100967  7.897884  7.699777  7.506548
##  [43]  7.318095  7.134315  6.955105  6.780359  6.609974  6.443848  6.281879
##  [50]  6.123966  5.970011  5.819917  5.673590  5.530935  5.391863  5.256284
##  [57]  5.124111  4.995259  4.869645  4.747189  4.627811  4.511434  4.397983
##  [64]  4.287384  4.179566  4.074459  3.971995  3.872108  3.774732  3.679805
##  [71]  3.587265  3.497052  3.409108  3.323375  3.239798  3.158323  3.078897
##  [78]  3.001469  2.925987  2.852404  2.780671  2.710742  2.642572  2.576116
##  [85]  2.511331  2.448176  2.386609  2.326590  2.268080  2.211042  2.155438
##  [92]  2.101233  2.048391  1.996877  1.946659  1.897704  1.849980  1.803457
##  [99]  1.758103  1.713890  1.670789
## 
## 
## [[69]]
## [[69]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[69]]$y
##   [1]  0.000000  4.390658  7.683377 10.126380 11.912399 13.191071 14.078551
##   [8] 14.664995 15.020356 15.198903 15.242719 15.184425 15.049291 14.856883
##  [15] 14.622332 14.357330 14.070900 13.769988 13.459935 13.144830 12.827796
##  [22] 12.511206 12.196850 11.886068 11.579851 11.278922 10.983796 10.694825
##  [29] 10.412240 10.136179  9.866703  9.603823  9.347505  9.097684  8.854273
##  [36]  8.617168  8.386251  8.161398  7.942476  7.729351  7.521886  7.319945
##  [43]  7.123389  6.932085  6.745896  6.564692  6.388342  6.216720  6.049701
##  [50]  5.887163  5.728986  5.575057  5.425260  5.279486  5.137627  4.999579
##  [57]  4.865238  4.734507  4.607288  4.483487  4.363013  4.245775  4.131687
##  [64]  4.020665  3.912626  3.807489  3.705178  3.605616  3.508730  3.414446
##  [71]  3.322697  3.233412  3.146527  3.061976  2.979698  2.899630  2.821714
##  [78]  2.745891  2.672106  2.600304  2.530431  2.462435  2.396267  2.331876
##  [85]  2.269216  2.208240  2.148902  2.091159  2.034967  1.980285  1.927073
##  [92]  1.875290  1.824899  1.775862  1.728143  1.681706  1.636516  1.592541
##  [99]  1.549748  1.508104  1.467580
## 
## 
## [[70]]
## [[70]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[70]]$y
##   [1]  0.000000  3.812707  6.753014  9.002510 10.705360 11.976007 12.905293
##   [8] 13.565319 14.013301 14.294636 14.445338 14.493967 14.463166 14.370880
##  [15] 14.231323 14.055748 13.853056 13.630282 13.392979 13.145526 12.891366
##  [22] 12.633205 12.373160 12.112882 11.853654 11.596467 11.342080 11.091067
##  [29] 10.843860 10.600775 10.362038 10.127802  9.898167  9.673188  9.452885
##  [36]  9.237252  9.026262  8.819873  8.618028  8.420665  8.227712  8.039094
##  [43]  7.854730  7.674539  7.498439  7.326345  7.158172  6.993839  6.833260
##  [50]  6.676355  6.523042  6.373241  6.226872  6.083860  5.944128  5.807602
##  [57]  5.674209  5.543878  5.416539  5.292123  5.170564  5.051796  4.935755
##  [64]  4.822380  4.711608  4.603380  4.497638  4.394325  4.293385  4.194763
##  [71]  4.098407  4.004264  3.912283  3.822416  3.734612  3.648825  3.565009
##  [78]  3.483119  3.403109  3.324937  3.248561  3.173939  3.101031  3.029798
##  [85]  2.960202  2.892204  2.825768  2.760858  2.697439  2.635477  2.574938
##  [92]  2.515789  2.458000  2.401538  2.346373  2.292475  2.239815  2.188365
##  [99]  2.138096  2.088983  2.040997
## 
## 
## [[71]]
## [[71]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[71]]$y
##   [1]  0.000000  4.291809  7.554078 10.012344 11.843182 13.184823 14.145474
##   [8] 14.809828 15.244165 15.500343 15.618932 15.631656 15.563320 15.433305
##  [15] 15.256747 15.045461 14.808660 14.553520 14.285622 14.009301 13.727916
##  [22] 13.444062 13.159736 12.876469 12.595428 12.317492 12.043322 11.773401
##  [29] 11.508080 11.247603 10.992133 10.741771 10.496567 10.256534 10.021656
##  [36]  9.791895  9.567198  9.347496  9.132715  8.922770  8.717574  8.517037
##  [43]  8.321066  8.129568  7.942447  7.759611  7.580965  7.406419  7.235881
##  [50]  7.069261  6.906471  6.747424  6.592036  6.440224  6.291905  6.147000
##  [57]  6.005430  5.867120  5.731994  5.599980  5.471005  5.345000  5.221898
##  [64]  5.101630  4.984131  4.869339  4.757191  4.647625  4.540583  4.436006
##  [71]  4.333837  4.234022  4.136505  4.041235  3.948158  3.857226  3.768387
##  [78]  3.681595  3.596802  3.513962  3.433029  3.353961  3.276713  3.201245
##  [85]  3.127515  3.055483  2.985110  2.916358  2.849190  2.783568  2.719458
##  [92]  2.656824  2.595633  2.535851  2.477446  2.420387  2.364641  2.310179
##  [99]  2.256972  2.204990  2.154206
## 
## 
## [[72]]
## [[72]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[72]]$y
##   [1]  0.000000  4.174364  7.335670  9.709219 11.470626 12.756763 13.674299
##   [8] 14.306357 14.717700 14.958782 15.068897 15.078642 15.011834 14.887002
##  [15] 14.718557 14.517699 14.293122 14.051571 13.798273 13.537266 13.271669
##  [22] 13.003880 12.735740 12.468651 12.203679 11.941626 11.683086 11.428499
##  [29] 11.178176 10.932338 10.691127 10.454632 10.222895  9.995927  9.773709
##  [36]  9.556208  9.343373  9.135141  8.931445  8.732207  8.537350  8.346790
##  [43]  8.160444  7.978229  7.800058  7.625847  7.455513  7.288972  7.126143
##  [50]  6.966944  6.811296  6.659122  6.510344  6.364888  6.222679  6.083646
##  [57]  5.947718  5.814827  5.684904  5.557883  5.433700  5.312291  5.193595
##  [64]  5.077551  4.964099  4.853182  4.744744  4.638728  4.535081  4.433750
##  [71]  4.334683  4.237829  4.143139  4.050566  3.960060  3.871577  3.785071
##  [78]  3.700498  3.617814  3.536978  3.457948  3.380684  3.305147  3.231297
##  [85]  3.159097  3.088510  3.019501  2.952034  2.886074  2.821587  2.758542
##  [92]  2.696906  2.636646  2.577733  2.520137  2.463827  2.408775  2.354954
##  [99]  2.302335  2.250892  2.200598
## 
## 
## [[73]]
## [[73]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[73]]$y
##   [1]  0.000000  4.153345  7.356232  9.806308 11.660532 13.043547 14.054318
##   [8] 14.771418 15.257218 15.561221 15.722706 15.772835 15.736317 15.632742
##  [15] 15.477630 15.283269 15.059384 14.813659 14.552164 14.279681 13.999976
##  [22] 13.716004 13.430079 13.144005 12.859184 12.576697 12.297372 12.021835
##  [29] 11.750555 11.483876 11.222042 10.965218 10.713511 10.466977 10.225637
##  [36]  9.989479  9.758473  9.532568  9.311701  9.095801  8.884785  8.678569
##  [43]  8.477064  8.280176  8.087814  7.899882  7.716287  7.536935  7.361732
##  [50]  7.190586  7.023408  6.860107  6.700595  6.544786  6.392595  6.243939
##  [57]  6.098737  5.956910  5.818379  5.683068  5.550902  5.421809  5.295718
##  [64]  5.172558  5.052262  4.934764  4.819998  4.707900  4.598410  4.491465
##  [71]  4.387008  4.284980  4.185325  4.087987  3.992913  3.900050  3.809347
##  [78]  3.720754  3.634220  3.549699  3.467144  3.386509  3.307749  3.230821
##  [85]  3.155682  3.082291  3.010606  2.940588  2.872199  2.805401  2.740156
##  [92]  2.676428  2.614183  2.553385  2.494001  2.435998  2.379344  2.324008
##  [99]  2.269958  2.217166  2.165602
## 
## 
## [[74]]
## [[74]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[74]]$y
##   [1]  0.000000  4.264760  7.541780 10.041497 11.929853 13.337698 14.368205
##   [8] 15.102717 15.605358 15.926673 16.106499 16.176224 16.160576 16.079026
##  [15] 15.946906 15.776278 15.576630 15.355414 15.118486 14.870434 14.614853
##  [22] 14.354553 14.091726 13.828075 13.564922 13.303283 13.043939 12.787484
##  [29] 12.534362 12.284904 12.039350 11.797867 11.560568 11.327521 11.098763
##  [36] 10.874300 10.654122 10.438201 10.226497 10.018961  9.815539  9.616169
##  [43]  9.420787  9.229327  9.041719  8.857895  8.677784  8.501317  8.328424
##  [50]  8.159035  7.993082  7.830497  7.671214  7.515166  7.362289  7.212519
##  [57]  7.065794  6.922051  6.781232  6.643276  6.508126  6.375725  6.246016
##  [64]  6.118946  5.994461  5.872509  5.753037  5.635995  5.521334  5.409006
##  [71]  5.298964  5.191160  5.085549  4.982086  4.880729  4.781433  4.684158
##  [78]  4.588861  4.495503  4.404045  4.314447  4.226672  4.140683  4.056443
##  [85]  3.973917  3.893070  3.813868  3.736277  3.660265  3.585799  3.512848
##  [92]  3.441381  3.371368  3.302779  3.235586  3.169760  3.105273  3.042098
##  [99]  2.980208  2.919578  2.860180
## 
## 
## [[75]]
## [[75]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[75]]$y
##   [1]  0.000000  4.226050  7.428320  9.831641 11.611997 12.907120 13.824794
##   [8] 14.449353 14.846772 15.068650 15.155333 15.138355 15.042353 14.886565
##  [15] 14.686000 14.452360 14.194759 13.920282 13.634431 13.341467 13.044682
##  [22] 12.746607 12.449184 12.153890 11.861841 11.573870 11.290591 11.012448
##  [29] 10.739750 10.472704 10.211437  9.956014  9.706451  9.462732  9.224808
##  [36]  8.992613  8.766065  8.545069  8.329522  8.119318  7.914344  7.714486
##  [43]  7.519629  7.329659  7.144459  6.963917  6.787921  6.616358  6.449122
##  [50]  6.286104  6.127201  5.972309  5.821329  5.674163  5.530714  5.390890
##  [57]  5.254600  5.121754  4.992266  4.866051  4.743026  4.623111  4.506228
##  [64]  4.392299  4.281251  4.173010  4.067505  3.964668  3.864431  3.766728
##  [71]  3.671495  3.578670  3.488191  3.400000  3.314039  3.230251  3.148582
##  [78]  3.068977  2.991385  2.915755  2.842036  2.770182  2.700144  2.631877
##  [85]  2.565336  2.500478  2.437259  2.375638  2.315576  2.257031  2.199968
##  [92]  2.144346  2.090131  2.037287  1.985779  1.935573  1.886637  1.838937
##  [99]  1.792444  1.747126  1.702954
## 
## 
## [[76]]
## [[76]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[76]]$y
##   [1]  0.000000  4.618916  8.097323 10.691598 12.601045 13.980587 14.950635
##   [8] 15.604764 16.015679 16.239863 16.321176 16.293673 16.183777 16.011983
##  [15] 15.794177 15.542657 15.266940 14.974373 14.670625 14.360054 14.046006
##  [22] 13.731036 13.417087 13.105628 12.797761 12.494300 12.195839 11.902803
##  [29] 11.615482 11.334066 11.058668 10.789339 10.526085 10.268879 10.017665
##  [36]  9.772368  9.532902  9.299166  9.071054  8.848454  8.631253  8.419332
##  [43]  8.212576  8.010867  7.814088  7.622124  7.434862  7.252190  7.073997
##  [50]  6.900176  6.730621  6.565228  6.403896  6.246526  6.093022  5.943288
##  [57]  5.797233  5.654766  5.515799  5.380248  5.248026  5.119054  4.993252
##  [64]  4.870540  4.750845  4.634090  4.520205  4.409119  4.300762  4.195069
##  [71]  4.091973  3.991410  3.893319  3.797638  3.704309  3.613274  3.524475
##  [78]  3.437859  3.353372  3.270961  3.190575  3.112165  3.035682  2.961078
##  [85]  2.888308  2.817326  2.748089  2.680553  2.614677  2.550419  2.487741
##  [92]  2.426604  2.366968  2.308799  2.252059  2.196713  2.142728  2.090069
##  [99]  2.038704  1.988602  1.939731
## 
## 
## [[77]]
## [[77]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[77]]$y
##   [1]  0.000000  4.860609  8.570990 11.378977 13.479521 15.025961 16.138897
##   [8] 16.913163 17.423326 17.728005 17.873268 17.895314 17.822574 17.677366
##  [15] 17.477205 17.235823 16.963977 16.670087 16.360730 16.041040 15.715013
##  [22] 15.385750 15.055651 14.726564 14.399903 14.076743 13.757894 13.443954
##  [29] 13.135360 12.832420 12.535342 12.244257 11.959236 11.680302 11.407444
##  [36] 11.140622 10.879774 10.624823 10.375683 10.132254  9.894435  9.662118
##  [43]  9.435192  9.213545  8.997067  8.785644  8.579165  8.377519  8.180598
##  [50]  7.988294  7.800502  7.617116  7.438037  7.263163  7.092396  6.925642
##  [57]  6.762807  6.603798  6.448527  6.296905  6.148847  6.004270  5.863092
##  [64]  5.725233  5.590615  5.459163  5.330800  5.205456  5.083059  4.963540
##  [71]  4.846831  4.732867  4.621581  4.512913  4.406800  4.303181  4.201999
##  [78]  4.103196  4.006717  3.912506  3.820510  3.730677  3.642956  3.557298
##  [85]  3.473655  3.391977  3.312221  3.234340  3.158290  3.084028  3.011512
##  [92]  2.940701  2.871556  2.804036  2.738104  2.673722  2.610854  2.549464
##  [99]  2.489518  2.430981  2.373821
## 
## 
## [[78]]
## [[78]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[78]]$y
##   [1]  0.000000  4.413785  7.763552 10.282278 12.152449 13.516988 14.487818
##   [8] 15.152579 15.579880 15.823422 15.925227 15.918164 15.827935 15.674627
##  [15] 15.473927 15.238076 14.976618 14.696982 14.404945 14.104985 13.800569
##  [22] 13.494370 13.188440 12.884346 12.583276 12.286121 11.993541 11.706019
##  [29] 11.423892 11.147394 10.876671 10.611803 10.352823 10.099723  9.852464
##  [36]  9.610989  9.375219  9.145068  8.920436  8.701220  8.487311  8.278598
##  [43]  8.074969  7.876310  7.682508  7.493451  7.309029  7.129130  6.953648
##  [50]  6.782477  6.615512  6.452652  6.293797  6.138850  5.987714  5.840298
##  [57]  5.696509  5.556259  5.419460  5.286030  5.155883  5.028941  4.905124
##  [64]  4.784354  4.666559  4.551663  4.439596  4.330288  4.223671  4.119679
##  [71]  4.018247  3.919313  3.822815  3.728692  3.636887  3.547343  3.460003
##  [78]  3.374813  3.291721  3.210675  3.131624  3.054519  2.979313  2.905959
##  [85]  2.834410  2.764624  2.696555  2.630162  2.565404  2.502241  2.440633
##  [92]  2.380541  2.321929  2.264760  2.208999  2.154611  2.101562  2.049818
##  [99]  1.999349  1.950123  1.902108
## 
## 
## [[79]]
## [[79]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[79]]$y
##   [1]  0.000000  4.647518  8.151246 10.767984 12.697408 14.094769 15.080757
##   [8] 15.749190 16.172981 16.408786 16.500619 16.482659 16.381439 16.217545
##  [15] 16.006940 15.761991 15.492268 15.205171 14.906409 14.600378 14.290455
##  [22] 13.979223 13.668652 13.360233 13.055086 12.754046 12.457722 12.166554
##  [29] 11.880848 11.600806 11.326552 11.058151 10.795619 10.538938 10.288063
##  [36] 10.042929  9.803456  9.569553  9.341123  9.118061  8.900260  8.687610
##  [43]  8.480001  8.277323  8.079464  7.886317  7.697772  7.513724  7.334068
##  [50]  7.158700  6.987520  6.820429  6.657331  6.498130  6.342735  6.191054
##  [57]  6.042999  5.898484  5.757424  5.619737  5.485343  5.354162  5.226118
##  [64]  5.101136  4.979142  4.860066  4.743838  4.630389  4.519653  4.411566
##  [71]  4.306063  4.203083  4.102566  4.004453  3.908687  3.815210  3.723969
##  [78]  3.634910  3.547981  3.463131  3.380310  3.299470  3.220563  3.143543
##  [85]  3.068365  2.994985  2.923360  2.853447  2.785207  2.718599  2.653583
##  [92]  2.590123  2.528180  2.467718  2.408702  2.351098  2.294872  2.239990
##  [99]  2.186420  2.134132  2.083094
## 
## 
## [[80]]
## [[80]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[80]]$y
##   [1]  0.000000  3.893024  6.892038  9.182783 10.912841 12.199459 13.135776
##   [8] 13.795774 14.238201 14.509706 14.647316 14.680416 14.632314 14.521491
##  [15] 14.362593 14.167217 13.944537 13.701807 13.444748 13.177872 12.904725
##  [22] 12.628088 12.350134 12.072556 11.796665 11.523468 11.253732 10.988036
##  [29] 10.726807 10.470355 10.218894  9.972569  9.731461  9.495612  9.265026
##  [36]  9.039678  8.819524  8.604505  8.394546  8.189567  7.989479  7.794188
##  [43]  7.603599  7.417613  7.236130  7.059052  6.886278  6.717710  6.553250
##  [50]  6.392802  6.236270  6.083562  5.934587  5.789253  5.647475  5.509164
##  [57]  5.374239  5.242615  5.114213  4.988955  4.866763  4.747563  4.631282
##  [64]  4.517848  4.407192  4.299246  4.193944  4.091221  3.991013  3.893260
##  [71]  3.797901  3.704878  3.614133  3.525610  3.439256  3.355017  3.272841
##  [78]  3.192678  3.114478  3.038194  2.963778  2.891185  2.820369  2.751289
##  [85]  2.683900  2.618162  2.554034  2.491477  2.430452  2.370922  2.312850
##  [92]  2.256200  2.200938  2.147029  2.094441  2.043141  1.993097  1.944279
##  [99]  1.896657  1.850201  1.804883
## 
## 
## [[81]]
## [[81]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[81]]$y
##   [1]  0.000000  4.051428  7.169205  9.549263 11.346812 12.684795 13.660589
##   [8] 14.351309 14.818019 15.109065 15.262713 15.309245 15.272616 15.171771
##  [15] 15.021682 14.834176 14.618587 14.382278 14.131046 13.869452 13.601076
##  [22] 13.328723 13.054583 12.780360 12.507375 12.236643 11.968941 11.704856
##  [29] 11.444825 11.189166 10.938107 10.691801 10.450343 10.213786  9.982146
##  [36]  9.755413  9.533555  9.316525  9.104263  8.896700  8.693762  8.495366
##  [43]  8.301431  8.111868  7.926592  7.745513  7.568545  7.395598  7.226587
##  [50]  7.061424  6.900026  6.742309  6.588190  6.437588  6.290425  6.146623
##  [57]  6.006106  5.868799  5.734629  5.603525  5.475418  5.350238  5.227920
##  [64]  5.108398  4.991608  4.877487  4.765976  4.657013  4.550542  4.446505
##  [71]  4.344846  4.245511  4.148448  4.053603  3.960927  3.870369  3.781882
##  [78]  3.695418  3.610931  3.528375  3.447707  3.368883  3.291861  3.216600
##  [85]  3.143060  3.071201  3.000985  2.932374  2.865332  2.799823  2.735811
##  [92]  2.673263  2.612145  2.552424  2.494069  2.437048  2.381330  2.326887
##  [99]  2.273688  2.221705  2.170911
## 
## 
## [[82]]
## [[82]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[82]]$y
##   [1]  0.000000  4.161174  7.374100  9.835478 11.701603 13.096654 14.119287
##   [8] 14.847868 15.344643 15.659040 15.830306 15.889593 15.861621 15.765998
##  [15] 15.618271 15.430759 15.213215 14.973359 14.717286 14.449811 14.174725
##  [22] 13.895007 13.612996 13.330516 13.048989 12.769514 12.492933 12.219891
##  [29] 11.950868 11.686221 11.426206 11.171001 10.920720 10.675431 10.435162
##  [36] 10.199912  9.969656  9.744352  9.523945  9.308368  9.097547  8.891403
##  [43]  8.689853  8.492809  8.300184  8.111889  7.927834  7.747931  7.572091
##  [50]  7.400225  7.232249  7.068075  6.907620  6.750801  6.597538  6.447750
##  [57]  6.301360  6.158291  6.018469  5.881819  5.748271  5.617754  5.490200
##  [64]  5.365541  5.243712  5.124649  5.008289  4.894571  4.783435  4.674822
##  [71]  4.568676  4.464939  4.363558  4.264478  4.167648  4.073017  3.980535
##  [78]  3.890152  3.801822  3.715497  3.631132  3.548683  3.468106  3.389359
##  [85]  3.312399  3.237187  3.163683  3.091848  3.021644  2.953034  2.885982
##  [92]  2.820452  2.756411  2.693823  2.632657  2.572879  2.514459  2.457365
##  [99]  2.401568  2.347037  2.293745
## 
## 
## [[83]]
## [[83]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[83]]$y
##   [1]  0.000000  4.171036  7.373296  9.810866 11.645291 13.004431 13.989475
##   [8] 14.680493 15.140835 15.420612 15.559454 15.588690 15.533081 15.412188
##  [15] 15.241452 15.033057 14.796605 14.539659 14.268165 13.986790 13.699192
##  [22] 13.408228 13.116122 12.824599 12.534989 12.248310 11.965334 11.686639
##  [29] 11.412650 11.143673 10.879918 10.621521 10.368561 10.121071  9.879050
##  [36]  9.642469  9.411279  9.185414  8.964798  8.749344  8.538960  8.333549
##  [43]  8.133012  7.937247  7.746152  7.559623  7.377560  7.199861  7.026425
##  [50]  6.857153  6.691949  6.530717  6.373363  6.219795  6.069923  5.923659
##  [57]  5.780917  5.641613  5.505664  5.372990  5.243512  5.117153  4.993838
##  [64]  4.873495  4.756052  4.641438  4.529586  4.420429  4.313903  4.209944
##  [71]  4.108490  4.009480  3.912857  3.818562  3.726540  3.636735  3.549094
##  [78]  3.463565  3.380097  3.298641  3.219148  3.141571  3.065863  2.991979
##  [85]  2.919876  2.849511  2.780841  2.713826  2.648426  2.584602  2.522316
##  [92]  2.461532  2.402212  2.344321  2.287826  2.232692  2.178887  2.126379
##  [99]  2.075135  2.025127  1.976324
## 
## 
## [[84]]
## [[84]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[84]]$y
##   [1]  0.000000  4.128308  7.340071  9.818711 11.711385 13.136140 14.187663
##   [8] 14.941893 15.459722 15.789971 15.971767 16.036466 16.009180 15.910018
##  [15] 15.755069 15.557201 15.326695 15.071760 14.798939 14.513442 14.219409
##  [22] 13.920123 13.618179 13.315621 13.014054 12.714727 12.418609 12.126442
##  [29] 11.838788 11.556065 11.278577 11.006534 10.740076 10.479287 10.224201
##  [36]  9.974821  9.731119  9.493046  9.260537  9.033514  8.811889  8.595567
##  [43]  8.384447  8.178425  7.977397  7.781254  7.589889  7.403194  7.221062
##  [50]  7.043388  6.870068  6.700997  6.536075  6.375202  6.218282  6.065217
##  [57]  5.915916  5.770285  5.628236  5.489682  5.354536  5.222716  5.094140
##  [64]  4.968728  4.846402  4.727088  4.610710  4.497197  4.386479  4.278485
##  [71]  4.173151  4.070409  3.970197  3.872452  3.777113  3.684122  3.593419
##  [78]  3.504950  3.418659  3.334492  3.252398  3.172324  3.094222  3.018043
##  [85]  2.943739  2.871265  2.800575  2.731625  2.664373  2.598776  2.534795
##  [92]  2.472388  2.411519  2.352147  2.294238  2.237754  2.182661  2.128924
##  [99]  2.076510  2.025387  1.975522
## 
## 
## [[85]]
## [[85]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[85]]$y
##   [1]  0.000000  4.371243  7.626844 10.025350 11.766004 13.002334 13.852626
##   [8] 14.407997 14.738624 14.898539 14.929331 14.862996 14.724139 14.531666
##  [15] 14.300095 14.040556 13.761576 13.469670 13.169811 12.865776 12.560428
##  [22] 12.255924 11.953878 11.655487 11.361628 11.072933 10.789843 10.512659
##  [29] 10.241569  9.976678  9.718030  9.465617  9.219398  8.979304  8.745248
##  [36]  8.517127  8.294829  8.078234  7.867219  7.661658  7.461422  7.266385
##  [43]  7.076419  6.891398  6.711200  6.535701  6.364781  6.198325  6.036215
##  [50]  5.878342  5.724594  5.574865  5.429050  5.287047  5.148757  5.014084
##  [57]  4.882933  4.755211  4.630830  4.509702  4.391742  4.276867  4.164997
##  [64]  4.056053  3.949959  3.846639  3.746022  3.648037  3.552615  3.459689
##  [71]  3.369193  3.281065  3.195242  3.111663  3.030271  2.951008  2.873818
##  [78]  2.798647  2.725443  2.654153  2.584728  2.517119  2.451278  2.387160
##  [85]  2.324718  2.263910  2.204693  2.147024  2.090864  2.036173  1.982913
##  [92]  1.931045  1.880535  1.831345  1.783443  1.736793  1.691363  1.647122
##  [99]  1.604038  1.562081  1.521221
## 
## 
## [[86]]
## [[86]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[86]]$y
##   [1]  0.000000  4.093195  7.214554  9.575407 11.341518 12.642898 13.581512
##   [8] 14.237320 14.673016 14.937750 15.070048 15.100099 15.051550 14.942922
##  [15] 14.788708 14.600248 14.386403 14.154095 13.908721 13.654484 13.394654
##  [22] 13.131763 12.867770 12.604184 12.342159 12.082576 11.826097 11.573216
##  [29] 11.324294 11.079592 10.839287 10.603497 10.372288 10.145691  9.923708
##  [36]  9.706318  9.493482  9.285151  9.081264  8.881753  8.686548  8.495571
##  [43]  8.308745  8.125989  7.947223  7.772366  7.601339  7.434060  7.270452
##  [50]  7.110435  6.953933  6.800871  6.651173  6.504767  6.361580  6.221544
##  [57]  6.084589  5.950647  5.819653  5.691542  5.566250  5.443716  5.323879
##  [64]  5.206679  5.092060  4.979963  4.870334  4.763119  4.658263  4.555716
##  [71]  4.455426  4.357344  4.261421  4.167610  4.075863  3.986137  3.898386
##  [78]  3.812566  3.728636  3.646553  3.566278  3.487769  3.410989  3.335899
##  [85]  3.262462  3.190642  3.120403  3.051710  2.984529  2.918827  2.854572
##  [92]  2.791731  2.730274  2.670169  2.611388  2.553900  2.497678  2.442694
##  [99]  2.388920  2.336330  2.284898
## 
## 
## [[87]]
## [[87]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[87]]$y
##   [1]  0.000000  4.359891  7.679881 10.185442 12.053650 13.423559 14.404355
##   [8] 15.081758 15.523059 15.781065 15.897213 15.904002 15.826916 15.685924
##  [15] 15.496669 15.271391 15.019661 14.748955 14.465101 14.172635 13.875081
##  [22] 13.575165 13.274991 12.976171 12.679936 12.387216 12.098705 11.814914
##  [29] 11.536209 11.262846 10.994993 10.732751 10.476166 10.225247  9.979969
##  [36]  9.740286  9.506132  9.277429  9.054088  8.836013  8.623105  8.415258
##  [43]  8.212368  8.014328  7.821030  7.632369  7.448238  7.268534  7.093152
##  [50]  6.921993  6.754956  6.591943  6.432860  6.277612  6.126108  5.978258
##  [57]  5.833975  5.693172  5.555766  5.421676  5.290822  5.163125  5.038509
##  [64]  4.916901  4.798228  4.682419  4.569405  4.459119  4.351494  4.246467
##  [71]  4.143974  4.043956  3.946351  3.851102  3.758152  3.667446  3.578928
##  [78]  3.492547  3.408251  3.325990  3.245714  3.167376  3.090928  3.016325
##  [85]  2.943523  2.872479  2.803149  2.735492  2.669468  2.605038  2.542163
##  [92]  2.480805  2.420929  2.362497  2.305476  2.249831  2.195529  2.142538
##  [99]  2.090826  2.040362  1.991115
## 
## 
## [[88]]
## [[88]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[88]]$y
##   [1]  0.000000  4.505042  7.901997 10.438441 12.307198 13.658435 14.609084
##   [8] 15.250186 15.652607 15.871502 15.949780 15.920815 15.810551 15.639146
##  [15] 15.422249 15.172000 14.897805 14.606942 14.305030 13.996399 13.684376
##  [22] 13.371506 13.059727 12.750505 12.444941 12.143847 11.847820 11.557282
##  [29] 11.272523 10.993734 10.721024 10.454444 10.193997  9.939654  9.691357
##  [36]  9.449028  9.212576  8.981900  8.756889  8.537428  8.323401  8.114686
##  [43]  7.911165  7.712716  7.519221  7.330561  7.146619  6.967281  6.792435
##  [50]  6.621969  6.455775  6.293749  6.135785  5.981784  5.831646  5.685274
##  [57]  5.542575  5.403457  5.267830  5.135607  5.006702  4.881032  4.758516
##  [64]  4.639076  4.522633  4.409112  4.298441  4.190548  4.085363  3.982818
##  [71]  3.882847  3.785386  3.690370  3.597740  3.507435  3.419396  3.333567
##  [78]  3.249893  3.168318  3.088792  3.011261  2.935677  2.861990  2.790152
##  [85]  2.720117  2.651841  2.585278  2.520386  2.457123  2.395448  2.335321
##  [92]  2.276703  2.219556  2.163844  2.109530  2.056580  2.004958  1.954633
##  [99]  1.905570  1.857739  1.811109
## 
## 
## [[89]]
## [[89]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[89]]$y
##   [1]  0.000000  3.934681  7.042570  9.481415 11.379175 12.839640 13.946987
##   [8] 14.769483 15.362485 15.770875 16.031036 16.172456 16.219029 16.190110
##  [15] 16.101373 15.965506 15.792772 15.591470 15.368305 15.128688 14.876984
##  [22] 14.616706 14.350677 14.081165 13.809981 13.538572 13.268087 12.999435
##  [29] 12.733329 12.470326 12.210854 11.955241 11.703730 11.456496 11.213665
##  [36] 10.975313 10.741488 10.512206 10.287462 10.067235  9.851488  9.640175
##  [43]  9.433241  9.230623  9.032256  8.838070  8.647992  8.461948  8.279862
##  [50]  8.101658  7.927261  7.756595  7.589584  7.426153  7.266229  7.109739
##  [57]  6.956611  6.806774  6.660160  6.516698  6.376324  6.238970  6.104573
##  [64]  5.973068  5.844396  5.718493  5.595302  5.474764  5.356822  5.241421
##  [71]  5.128505  5.018021  4.909917  4.804142  4.700645  4.599378  4.500292
##  [78]  4.403341  4.308478  4.215659  4.124839  4.035976  3.949028  3.863952
##  [85]  3.780710  3.699260  3.619566  3.541588  3.465290  3.390636  3.317590
##  [92]  3.246118  3.176185  3.107759  3.040807  2.975298  2.911200  2.848483
##  [99]  2.787117  2.727073  2.668322
## 
## 
## [[90]]
## [[90]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[90]]$y
##   [1]  0.000000  4.303900  7.552199  9.980261 11.771495 13.068812 13.983557
##   [8] 14.602467 14.993095 15.208047 15.288274 15.265644 15.164944 15.005447
##  [15] 14.802124 14.566597 14.307877 14.032939 13.747178 13.454750 13.158855
##  [22] 12.861942 12.565879 12.272082 11.981614 11.695265 11.413615 11.137076
##  [29] 10.865936 10.600383 10.340530 10.086431  9.838097  9.595502  9.358598
##  [36]  9.127316  8.901573  8.681274  8.466321  8.256605  8.052020  7.852452
##  [43]  7.657791  7.467925  7.282742  7.102132  6.925987  6.754198  6.586662
##  [50]  6.423275  6.263935  6.108544  5.957004  5.809221  5.665102  5.524558
##  [57]  5.387498  5.253838  5.123493  4.996382  4.872424  4.751540  4.633656
##  [64]  4.518696  4.406587  4.297261  4.190646  4.086676  3.985286  3.886412
##  [71]  3.789990  3.695960  3.604264  3.514842  3.427639  3.342599  3.259669
##  [78]  3.178797  3.099931  3.023022  2.948021  2.874881  2.803555  2.733999
##  [85]  2.666168  2.600021  2.535514  2.472608  2.411263  2.351439  2.293100
##  [92]  2.236208  2.180728  2.126624  2.073863  2.022410  1.972234  1.923303
##  [99]  1.875586  1.829053  1.783674
## 
## 
## [[91]]
## [[91]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[91]]$y
##   [1]  0.000000  4.531245  7.935226 10.469477 12.333163 13.680294 14.629958
##   [8] 15.274226 15.684274 15.915113 16.009254 15.999537 15.911324 15.764192
##  [15] 15.573244 15.350124 15.103802 14.841176 14.567548 14.286982 14.002587
##  [22] 13.716734 13.431222 13.147413 12.866324 12.588714 12.315138 12.045996
##  [29] 11.781567 11.522039 11.267529 11.018100 10.773776 10.534545 10.300378
##  [36] 10.071221  9.847014  9.627682  9.413146  9.203322  8.998121  8.797454
##  [43]  8.601230  8.409357  8.221746  8.038306  7.858947  7.683581  7.512121
##  [50]  7.344483  7.180581  7.020334  6.863660  6.710481  6.560719  6.414298
##  [57]  6.271145  6.131185  5.994348  5.860566  5.729768  5.601889  5.476865
##  [64]  5.354630  5.235124  5.118284  5.004052  4.892370  4.783180  4.676427
##  [71]  4.572056  4.470015  4.370252  4.272715  4.177354  4.084122  3.992971
##  [78]  3.903854  3.816726  3.731543  3.648261  3.566837  3.487231  3.409401
##  [85]  3.333309  3.258915  3.186181  3.115070  3.045547  2.977575  2.911120
##  [92]  2.846148  2.782627  2.720523  2.659805  2.600442  2.542404  2.485662
##  [99]  2.430186  2.375948  2.322920
## 
## 
## [[92]]
## [[92]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[92]]$y
##   [1]  0.000000  4.077824  7.193663  9.555180 11.325599 12.633201 13.578800
##   [8] 14.241615 14.683887 14.954510 15.091887 15.126173 15.081038 14.975056
##  [15] 14.822795 14.635673 14.422634 14.190675 13.945268 13.690682 13.430241
##  [22] 13.166530 12.901552 12.636851 12.373616 12.112750 11.854941 11.600698
##  [29] 11.350400 11.104318 10.862640 10.625491 10.392944 10.165034  9.941767
##  [36]  9.723125  9.509071  9.299558  9.094525  8.893906  8.697628  8.505617
##  [43]  8.317792  8.134074  7.954382  7.778635  7.606751  7.438649  7.274250
##  [50]  7.113474  6.956245  6.802485  6.652119  6.505073  6.361274  6.220652
##  [57]  6.083137  5.948661  5.817156  5.688557  5.562801  5.439824  5.319565
##  [64]  5.201965  5.086964  4.974505  4.864532  4.756991  4.651826  4.548987
##  [71]  4.448421  4.350078  4.253909  4.159866  4.067903  3.977972  3.890029
##  [78]  3.804031  3.719934  3.637696  3.557276  3.478634  3.401730  3.326527
##  [85]  3.252986  3.181071  3.110745  3.041975  2.974725  2.908961  2.844652
##  [92]  2.781764  2.720266  2.660128  2.601320  2.543811  2.487574  2.432580
##  [99]  2.378802  2.326213  2.274787
## 
## 
## [[93]]
## [[93]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[93]]$y
##   [1]  0.000000  4.362968  7.731661 10.312352 12.268934 13.731621 14.803856
##   [8] 15.567810 16.088744 16.418483 16.598174 16.660475 16.631302 16.531211
##  [15] 16.376500 16.180080 15.952179 15.700883 15.432587 15.152334 14.864098
##  [22] 14.571002 14.275498 13.979497 13.684489 13.391625 13.101791 12.815660
##  [29] 12.533737 12.256395 11.983903 11.716446 11.454147 11.197074 10.945259
##  [36] 10.698699 10.457369 10.221226  9.990211  9.764254  9.543280  9.327203
##  [43]  9.115937  8.909392  8.707475  8.510093  8.317153  8.128561  7.944226
##  [50]  7.764054  7.587955  7.415840  7.247621  7.083211  6.922526  6.765482
##  [57]  6.611997  6.461991  6.315387  6.172107  6.032077  5.895222  5.761471
##  [64]  5.630755  5.503003  5.378150  5.256129  5.136876  5.020328  4.906425
##  [71]  4.795106  4.686312  4.579987  4.476074  4.374518  4.275267  4.178268
##  [78]  4.083469  3.990821  3.900275  3.811783  3.725299  3.640778  3.558174
##  [85]  3.477444  3.398546  3.321438  3.246079  3.172430  3.100452  3.030107
##  [92]  2.961359  2.894170  2.828505  2.764330  2.701612  2.640316  2.580411
##  [99]  2.521865  2.464648  2.408728
## 
## 
## [[94]]
## [[94]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[94]]$y
##   [1]  0.000000  4.807979  8.459016 11.208137 13.254623 14.754187 15.828488
##   [8] 16.572568 17.060665 17.350761 17.488128 17.508108 17.438278 17.300154
##  [15] 17.110507 16.882408 16.626030 16.349288 16.058327 15.757914 15.451738
##  [22] 15.142646 14.832827 14.523959 14.217319 13.913872 13.614339 13.319255
##  [29] 13.029004 12.743858 12.463999 12.189541 11.920544 11.657029 11.398983
##  [36] 11.146372 10.899142 10.657226 10.420546 10.189020  9.962556  9.741062
##  [43]  9.524444  9.312604  9.105445  8.902871  8.704785  8.511092  8.321698
##  [50]  8.136509  7.955435  7.778385  7.605271  7.436006  7.270506  7.108688
##  [57]  6.950469  6.795771  6.644515  6.496625  6.352026  6.210645  6.072410
##  [64]  5.937252  5.805102  5.675893  5.549560  5.426039  5.305267  5.187183
##  [71]  5.071727  4.958841  4.848468  4.740551  4.635036  4.531870  4.431000
##  [78]  4.332375  4.235945  4.141662  4.049477  3.959344  3.871217  3.785052
##  [85]  3.700805  3.618433  3.537894  3.459148  3.382154  3.306874  3.233270
##  [92]  3.161304  3.090940  3.022142  2.954876  2.889106  2.824801  2.761926
##  [99]  2.700452  2.640345  2.581577
## 
## 
## [[95]]
## [[95]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[95]]$y
##   [1]  0.000000  4.203991  7.400834  9.807847 11.596026 12.899905 13.825314
##   [8] 14.455491 14.855890 15.077966 15.162158 15.140233 15.037132 14.872429
##  [15] 14.661469 14.416273 14.146245 13.858732 13.559461 13.252889 12.942471
##  [22] 12.630875 12.320154 12.011873 11.707220 11.407082 11.112114 10.822789
##  [29] 10.539436 10.262274  9.991435  9.726983  9.468933  9.217258  8.971901
##  [36]  8.732785  8.499811  8.272873  8.051851  7.836623  7.627059  7.423030
##  [43]  7.224404  7.031051  6.842838  6.659637  6.481320  6.307761  6.138836
##  [50]  5.974425  5.814410  5.658673  5.507104  5.359590  5.216024  5.076301
##  [57]  4.940320  4.807979  4.679183  4.553835  4.431845  4.313122  4.197579
##  [64]  4.085131  3.975695  3.869191  3.765539  3.664664  3.566492  3.470949
##  [71]  3.377965  3.287473  3.199404  3.113695  3.030282  2.949104  2.870100
##  [78]  2.793212  2.718385  2.645561  2.574689  2.505715  2.438589  2.373262
##  [85]  2.309684  2.247810  2.187593  2.128989  2.071955  2.016449  1.962431
##  [92]  1.909859  1.858695  1.808902  1.760443  1.713283  1.667385  1.622718
##  [99]  1.579246  1.536940  1.495766
## 
## 
## [[96]]
## [[96]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[96]]$y
##   [1]  0.000000  4.795173  8.412078 11.114727 13.108526 14.553273 15.573275
##   [8] 16.265211 16.704268 16.948901 17.044549 17.026517 16.922231 16.752981
##  [15] 16.535284 16.281946 16.002882 15.705762 15.396507 15.079680 14.758786
##  [22] 14.436511 14.114902 13.795509 13.479499 13.167740 12.860868 12.559343
##  [29] 12.263482 11.973498 11.689521 11.411618 11.139807 10.874070 10.614361
##  [36] 10.360612 10.112741  9.870653  9.634246  9.403412  9.178039  8.958015
##  [43]  8.743223  8.533549  8.328877  8.129095  7.934089  7.743750  7.557967
##  [50]  7.376635  7.199647  7.026901  6.858297  6.693735  6.533120  6.376358
##  [57]  6.223355  6.074023  5.928274  5.786021  5.647182  5.511673  5.379416
##  [64]  5.250332  5.124346  5.001382  4.881369  4.764236  4.649913  4.538334
##  [71]  4.429432  4.323144  4.219405  4.118157  4.019337  3.922889  3.828755
##  [78]  3.736880  3.647210  3.559692  3.474273  3.390904  3.309536  3.230120
##  [85]  3.152610  3.076960  3.003125  2.931062  2.860728  2.792082  2.725083
##  [92]  2.659692  2.595870  2.533580  2.472784  2.413447  2.355534  2.299010
##  [99]  2.243843  2.190000  2.137448
## 
## 
## [[97]]
## [[97]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[97]]$y
##   [1]  0.000000  4.664848  8.184421 10.815680 12.758444 14.168086 15.165408
##   [8] 15.844320 16.277819 16.522630 16.622830 16.612650 16.518670 16.361516
##  [15] 16.157187 15.918081 15.653796 15.371755 15.077690 14.776017 14.470131
##  [22] 14.162632 13.855504 13.550252 13.248011 12.949626 12.655719 12.366740
##  [29] 12.083006 11.804728 11.532040 11.265014 11.003677 10.748017 10.497997
##  [36] 10.253561 10.014635  9.781136  9.552971  9.330044  9.112252  8.899493
##  [43]  8.691661  8.488651  8.290359  8.096680  7.907512  7.722751  7.542298
##  [50]  7.366056  7.193926  7.025814  6.861627  6.701275  6.544668  6.391720
##  [57]  6.242345  6.096459  5.953983  5.814835  5.678939  5.546219  5.416600
##  [64]  5.290010  5.166379  5.045637  4.927716  4.812552  4.700078  4.590234
##  [71]  4.482956  4.378186  4.275864  4.175933  4.078338  3.983024  3.889938
##  [78]  3.799026  3.710240  3.623528  3.538844  3.456138  3.375365  3.296480
##  [85]  3.219438  3.144197  3.070715  2.998949  2.928861  2.860411  2.793561
##  [92]  2.728273  2.664511  2.602239  2.541423  2.482028  2.424020  2.367369
##  [99]  2.312042  2.258007  2.205236
## 
## 
## [[98]]
## [[98]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[98]]$y
##   [1]  0.000000  4.537610  7.978770 10.566672 12.491000 13.899691 14.908118
##   [8] 15.606247 16.064224 16.336731 16.466389 16.486410 16.422665 16.295298
##  [15] 16.119988 15.908931 15.671604 15.415367 15.145927 14.867703 14.584108
##  [22] 14.297774 14.010723 13.724499 13.440279 13.158951 12.881179 12.607451
##  [29] 12.338121 12.073440 11.813575 11.558630 11.308663 11.063693 10.823710
##  [36] 10.588684 10.358567 10.133299  9.912812  9.697031  9.485875  9.279260
##  [43]  9.077103  8.879315  8.685810  8.496502  8.311303  8.130129  7.952894
##  [50]  7.779515  7.609909  7.443997  7.281698  7.122935  6.967632  6.815712
##  [57]  6.667104  6.521735  6.379534  6.240434  6.104366  5.971265  5.841065
##  [64]  5.713704  5.589120  5.467252  5.348042  5.231430  5.117361  5.005780
##  [71]  4.896631  4.789862  4.685422  4.583258  4.483322  4.385565  4.289940
##  [78]  4.196400  4.104899  4.015394  3.927840  3.842195  3.758418  3.676467
##  [85]  3.596303  3.517887  3.441181  3.366148  3.292751  3.220954  3.150722
##  [92]  3.082022  3.014820  2.949083  2.884779  2.821878  2.760348  2.700160
##  [99]  2.641284  2.583692  2.527356
## 
## 
## [[99]]
## [[99]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[99]]$y
##   [1]  0.000000  3.921301  6.949424  9.269931 11.030192 12.347233 13.313982
##   [8] 14.004222 14.476525 14.777385 14.943698 15.004739 14.983727 14.899075
##  [15] 14.765373 14.594180 14.394646 14.174006 13.937977 13.691072 13.436844
##  [22] 13.178086 12.916991 12.655270 12.394257 12.134983 11.878244 11.624644
##  [29] 11.374639 11.128566 10.886671 10.649123 10.416035 10.187476  9.963476
##  [36]  9.744040  9.529148  9.318768  9.112851  8.911339  8.714170  8.521272
##  [43]  8.332572  8.147993  7.967457  7.790885  7.618198  7.449315  7.284158
##  [50]  7.122648  6.964708  6.810261  6.659232  6.511547  6.367132  6.225916
##  [57]  6.087830  5.952804  5.820771  5.691665  5.565421  5.441977  5.321270
##  [64]  5.203239  5.087827  4.974974  4.864623  4.756721  4.651211  4.548042
##  [71]  4.447161  4.348517  4.252062  4.157746  4.065522  3.975344  3.887165
##  [78]  3.800943  3.716633  3.634194  3.553582  3.474759  3.397685  3.322320
##  [85]  3.248626  3.176568  3.106107  3.037210  2.969840  2.903965  2.839552
##  [92]  2.776567  2.714979  2.654757  2.595871  2.538291  2.481989  2.426935
##  [99]  2.373102  2.320464  2.268993
## 
## 
## [[100]]
## [[100]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[100]]$y
##   [1]  0.000000  4.070061  7.176210  9.527079 11.286545 12.583319 13.518486
##   [8] 14.171425 14.604460 14.866519 14.996003 15.023045 14.971281 14.859244
##  [15] 14.701459 14.509300 14.291671 14.055531 13.806315 13.548260 13.284662
##  [22] 13.018080 12.750493 12.483427 12.218050 11.955252 11.695702 11.439901
##  [29] 11.188214 10.940901 10.698142 10.460052 10.226697  9.998106  9.774277
##  [36]  9.555186  9.340792  9.131039  8.925865  8.725196  8.528957  8.337068
##  [43]  8.149445  7.966005  7.786664  7.611335  7.439935  7.272379  7.108585
##  [50]  6.948471  6.791956  6.638961  6.489408  6.343220  6.200322  6.060642
##  [57]  5.924106  5.790645  5.660190  5.532672  5.408027  5.286190  5.167097
##  [64]  5.050686  4.936898  4.825674  4.716955  4.610685  4.506809  4.405274
##  [71]  4.306026  4.209014  4.114188  4.021497  3.930896  3.842335  3.755769
##  [78]  3.671154  3.588445  3.507600  3.428576  3.351332  3.275828  3.202026
##  [85]  3.129886  3.059372  2.990446  2.923073  2.857218  2.792846  2.729925
##  [92]  2.668422  2.608304  2.549540  2.492101  2.435955  2.381074  2.327430
##  [99]  2.274995  2.223740  2.173641

Example 2: multi-factor experiments to build (hierarchical) logistic regression models for pharmaceutical salt formation

Four controllable variables:

• rate of agitation during mixing (\(x_1\))
• volume of composition (\(x_2\))
• temperature (\(x_3\))
• evaporation rate (\(x_4\))

For the \(j\)th observation in the \(i\)th group \((i=1,\ldots,g;\, j=1,\ldots,n_g)\): \[ y_{ij} \sim \mbox{Bernoulli}\left(\rho(\boldsymbol{x}_{ij})\right) \] with \[ \log\left(\frac{\rho(\boldsymbol{x}_{ij})}{1-\rho(\boldsymbol{x}_{ij})}\right) = \left(\beta_0 + \omega_{i0}\right) + \sum_{r=1}^k\left(\beta_r + \omega_{ir}\right)x_{ijr}\,, \] where \(x_{ijr}\) is the value taken by the \(r\)th variable.

• \(\boldsymbol{\beta}= (\beta_0,\beta_1,\ldots,\beta_{q-1})^\mathrm{T}\) are unknown parameters of interest
• \(\boldsymbol{\omega}_i = (\omega_{i0}, \omega_{i1}, \ldots, \omega_{iq-1})^\mathrm{T}\) are group specific parameters for the \(i\)th group

Prior distributions (for later use):

• \(\beta_0 \sim U(-3,3)\), \(\beta_1 \sim U(4, 10)\), \(\beta_2 \sim U(5, 11)\), \(\beta_3 \sim U(-6, 0)\), \(\beta_4 \sim U(-2.5, 3.5)\)

1. standard logistic regression - \(\omega_{ir} = 0\)
2. hierarchical logistic regression - \(\omega_{ir} \sim U(-s_r, +s_r)\). with \(s_{r}>0\) following a triangular distribution

Many Frequentist criteria for finding optimal designs for both linear and nonlinear models optimise a function of the information matrix; see Atkinson, Donev, and Tobias (2007), ch.10

• we have already seen \(D\)-optimality

Let \(\xi = (\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n)^\mathrm{T}\) denote a design, and set \(M(\xi;\,\boldsymbol{\theta}) = M(\boldsymbol{\theta})\) to explicitly acknowledge the dependence of the information matrix on the design

• \(D\)-optimality: maximise \(\phi_D(\xi) = \mbox{det}\, M(\xi;\,\boldsymbol{\theta})\)
• \(A\)-optimality: minimise \(\phi_A(\xi) = \mbox{trace}\, M(\xi;\,\boldsymbol{\theta})^{-1}\)
• \(G\)-optimality: minimise \(\phi_G(\xi) = \max_\boldsymbol{x}\mbox{Var}(\hat{y}(\boldsymbol{x}))\)
  • where \(\hat{y}(x)\) is the predicted response at \(\boldsymbol{x}\) and the (asymptotic) prediction variance is a function of \(M(\xi;\,\boldsymbol{\theta})\)
• \(V\)- (or \(I\)-) optimality - minimise \(\phi_V(\xi) = \int_\mathcal{X} \mbox{Var}\left(\hat{y}(\boldsymbol{x})\right)\,\mathrm{d}\boldsymbol{x}\)

For most nonlinear models, \(M(\xi;\,\boldsymbol{\theta})\) will be a function of the unknown parameters \(\boldsymbol{\theta}\) (unlike for the linear model, where \(M(\xi;\,\boldsymbol{\beta}) = X^\mathrm{T}X / \sigma^2\))

This leads to a “chicken and egg” situation

• if you can tell me the values of the unknown parameters, I can give you an optimal design
• but if you knew the value of \(\boldsymbol{\theta}\), you probably wouldn’t need to perform the experiment!

For some models/experiments, the quality of a design may change a lot with the value of \(\boldsymbol{\theta}\)

rho <- function(x, beta0 = 0, beta1 = 1) {
  eta <- beta0 + beta1 * x
  1 / (1 + exp(-eta))
}
par(mar = c(8, 4, 1, 2) + 0.1)
curve(rho, from = -5, to = 5, ylab = expression(rho), xlab = expression(italic(x)), cex.lab = 1.5, 
      cex.axis = 1.5, ylim = c(0, 1), lwd = 2)

For simple logistic regression, the information matrix has the form \[ M(\xi;\,\boldsymbol{\beta}) = X^\mathrm{T}W X\,, \] with \(X\) the \(n\times 2\) model matrix and \(W = \mbox{diag}\left\{\rho(x_i)[1-\rho(x_i)]\right\}\)

For example with \(n=2\), \(\xi = (-1, 1)\), \(\beta_0=0\) and \(\beta_1 = 1\) \[ M(\xi;\,\boldsymbol{\beta}) = \left( \begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array} \right) \left( \begin{array}{cc} 0.2 & 0 \\ 0 & 0.2 \end{array} \right) \left( \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right) \]

Minfo <- function(xi, beta0 = 0, beta1 = 1) {
  X <- cbind(c(1, 1), xi)
  v <- function(x) rho(x, beta0, beta1) * (1 - rho(x, beta0, beta1))
  W <- diag(c(v(xi[1]), v(xi[2])))
  t(X) %*% W %*% X
}
Dcrit <- function(xi, beta0 = 0, beta1 = 1) {
  d <- det(Minfo(xi, beta0, beta1))
  ifelse(is.nan(d), -Inf, d)
}

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 1\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1))
xi.opt1 <- dopt$par
xi.opt1

## [1] -1.543421  1.543530

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 2\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1), beta1 = 2)
xi.opt2 <- dopt$par
xi.opt2

## [1] -0.7717705  0.7717418

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 0.5\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1), beta1 = .5)
xi.opt3 <- dopt$par
xi.opt3

## [1] -3.086913  3.086981

Getting \(\beta_1\) wrong: design for \(\beta_1 = .5\) when actually \(\beta_1 = 2\)

\(D\)-efficiency

(Dcrit(xi.opt3, beta1 = 2) / Dcrit(xi.opt2, beta1 = 2)) ^ (1 / 2)

## [1] 0.05720905

Use of the “wrong” design can lead to uninformative experiments (with “small” information matrices)

For the logistic regression example, the drop in efficiency is closely related to the phenomenon of separation (see Firth 1993)

Motivates the need for designs which are robust to the values of the model parameters

• maximin designs (focus on worst case performance)
• Bayesian designs

Decision-theoretic design starts with a utility function \(u(\xi,\boldsymbol{y},\boldsymbol{\theta})\) that defines the usefulness of a design for a particular purpose, given data \(\boldsymbol{y}\) and parameters \(\boldsymbol{\theta}\)

Common choices of utility function include

• negative squared error loss \[u(\xi, \boldsymbol{y}, \boldsymbol{\theta}) = -\left[\boldsymbol{\theta}- E(\boldsymbol{\theta}\,|\,\boldsymbol{y})\right]^2\]
  • negative squared difference between \(\boldsymbol{\theta}\) and the posterior mean
• surprisal or self information \[ \begin{split} u(\xi, \boldsymbol{y}, \boldsymbol{\theta}) & = \log \pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi) - \log \pi(\boldsymbol{\theta}) \\ & = \log \pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi) - \log \pi(\boldsymbol{y}\,|\,\xi) \end{split} \]
  • difference between log posterior and log prior densities, or between the log-likelihood and the log-evidence

A priori (before the experiment), we do not know \(\boldsymbol{y}\) or \(\boldsymbol{\theta}\) (we will never know \(\boldsymbol{\theta}\))

So, we take the expectation of the utility function with respect to the joint distribution of \(\boldsymbol{y},\boldsymbol{\theta}\)

\[ \begin{split} U(\xi) & = E_{\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi}\left[u(\xi,\boldsymbol{y},\boldsymbol{\theta})\right]\\ & = \int u(\xi,\boldsymbol{y},\boldsymbol{\theta})\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int u(\xi, \boldsymbol{y}, \boldsymbol{\theta})\pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)\pi(\boldsymbol{y}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int u(\xi, \boldsymbol{y}, \boldsymbol{\theta})\pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi)\pi(\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y} \end{split} \] The equivalence of the third and fourth equations follows from Bayes theorem

• the third equation more clearly shows the dependence on the posterior distribution
• the fourth equation is often more useful for calculations and computation

See Chaloner and Verdinelli (1995)

Surprisal \[ \begin{split} U(\xi) & = \int \log \frac{\pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)}{\pi(\boldsymbol{\theta})}\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int \log \frac{\pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi)}{\pi(\boldsymbol{y}\,|\,\xi)}\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y} \end{split} \] - the expected Shannon information gain (SIG) or expected Kullback-Liebler divergence between prior and posterior densities

Negative squared error loss \[ \begin{split} U(\xi) & = - \int \left[\boldsymbol{\theta}- E(\boldsymbol{\theta}\,|\,\boldsymbol{y})\right]^2\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = - \int \mbox{tr}\left\{\mbox{Var}(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)\pi(\boldsymbol{y}\,|\,\xi)\right\}\,\mathrm{d}\boldsymbol{y} \end{split} \] - the expected negative squared error loss (NSEL)

In general, Bayesian design is easy in principle but hard in practice

1. For most nonlinear models, the expected utility will be intractable and involves high-dimensional integrals with respect to \(\boldsymbol{y}\) - often, obtaining the utility function itself requires the solution of intractable integrals (cf both ESIG and NSEL) - numerical or analytical approximation is required (eg Ryan et al. 2016)

2. A high-dimensional optimisation problem results for multi-factor experiments with many design points

For large \(n\), the inverse information matrix \(M(\xi;\,\boldsymbol{\theta})\) is an asymptotic approximation to the posterior variance-covariance matrix

Using this approximation, we can define Bayesian analogues of classical optimality criteria

\(D\)-optimality: maximise \[ U_D(\xi) = \int \log\mbox{det} M(\xi;\,\boldsymbol{\theta})\pi(\boldsymbol{\theta})\,\mathrm{d}\boldsymbol{\theta} \]

• approximation to ESIG

\(A\)-optimality: maximise \[ U_A(\xi) = - \int \mbox{tr} M^{-1}(\xi;\,\boldsymbol{\theta})\pi(\boldsymbol{\theta})\,\mathrm{d}\boldsymbol{\theta} \]

• approximation to NSEL

These integrals, with respect to \(\boldsymbol{\theta}\), are lower dimensional and more amenable to deterministic (quadrature) approximation, eg Gotwalt, Jones, and Steinberg (2009)

The acebayes package provides functions for constructing approximations to expected utilities

• default is to use quadrature to approximate the Bayesian \(D\)-optimality objective function

library(acebayes)
prior <- list(support = matrix(c(0, 0, .5, 2), nrow = 2))
logreg.util <- utilityglm(formula = ~ x, family = binomial, prior = prior)$utility
BDcrit <- function(xi) logreg.util(data.frame(x = xi))
bdopt <- optim(par = c(-1, 1), BDcrit, control = list(fnscale = -1))
bdopt$par

## [1] -1.202960  1.203132

As an alternative to analytical approximations, Monte Carlo approximation to the expected utility is simple to implement and intuitively appealing

\[ \tilde{U}(\xi) = \frac{1}{B}\sum_{i=1}^B\tilde{u}(\xi, \boldsymbol{y}_i, \boldsymbol{\theta}_i) \] where

• \(\left\{\boldsymbol{\theta}_h, \boldsymbol{y}_h\right\}_{h=1}^B\) is a random sample from \(\pi(\boldsymbol{\theta},\boldsymbol{y}\,|\,\xi)\)
• \(\tilde{u}(\xi,\boldsymbol{y},\boldsymbol{\theta})\) is, where necessary, an approximation to the utility function (often, nested Monte Carlo is required)

How to construct the approximation \(\tilde{u}(\xi,\boldsymbol{y},\boldsymbol{\theta})\) is an active area of research, eg Overstall, McGree, and Drovandi (2018), Beck et al. (2018)

Find an optimal design using Monte Carlo:

priorMC <- function(B) cbind(rep(0, B), runif(n = B, min = .5, max = 2))
logreg.utilSIG <- utilityglm(formula = ~ x, family = binomial, prior = priorMC, criterion = "SIG")$utility
BDcritSIG <- function(xi, B = 1000) mean(logreg.utilSIG(data.frame(x = xi), B))
bdoptSIG <- optim(par = c(-1, 1), BDcritSIG, control = list(fnscale = -1))
bdoptSIG$par

## [1] -1.015625  1.079688

bdopt$par

## [1] -1.202960  1.203132

Larger Monte Carlo sample sizes will produce results more similar to the design found using quadrature (in this example)

In general, direct optimisation of the Monte Carlo approximation requires large \(B\) to generate suitable smooth objective function and/or expensive stochastic algorithms (eg genetic algorithms)

Hamada et al. (2001)

Alternatively, the optimisation can be embedded within a simulation scheme and samples generated from the joint artificial distribution of \(\xi,\boldsymbol{y},\boldsymbol{\theta}\)

• take \(\xi^*\), the optimal design, to be the posterior mode of the marginal distribution
• most effective for small experiments (both numbers of variables and runs)

Müller (1999), Müller, Sansó, and De Iorio (2004)

Instead of directly minimising a Monte Carlo approximation to the expected utility, find designs via curve fitting (Müller and Parmigiani 1996)

1. Evaluate the Monte Carlo approximation \(\tilde{U}(\xi)\) for a small number of designs, \(\xi_1,\ldots,\xi_Q\)
2. Smooth the “data” \(\left\{\xi_i, \tilde{U}(\xi_i)\right\}\), i.e. fit a statistical model, to obtain a surrogate \(\hat{U}(\xi)\)
3. Find \(\xi\) that maximises \(\hat{U}(\xi)\)

Return to Example 1, compartmental model

• find a design with \(n=2\) runs, with fixed \(x_1 = 5\)
• use Monte Carlo approximation to SIG for 10 values of \(x_2\)

library(DiceKriging)
library(DiceDesign)
n <- 10; x1<- -0.583; x2 <- 2 * maximinSA_LHS(lhsDesign(n, 1)$design)$design- 1
u <- NULL; for(i in 1:n) u[i] <- mean(utilcomp15sig(c(x1, x2[i]), B = 1000))
par(mar = c(4, 4, 2, 2) + 0.1)
plot(12 * (x2 + 1), u, xlab = expression(x[2]), ylab = "Approx. expected SIG", xlim = c(0, 24), 
     ylim = c(0, 2), pch = 16, cex = 1.5); abline(v = 12 * (x1 + 1), lwd = 2)
usmooth <- km(design = 12 * (x2 + 1), response = u, nugget = 1e-3, control = list(trace = F))
xgrid <- matrix(seq(0, 24, l = 1000), ncol = 1); pred <- predict(usmooth, xgrid, type = "SK")$mean
lines(seq(0, 24, l = 1000), pred, col = "blue", lwd = 2); abline(v = xgrid[which.max(pred), ], lty = 2)

Coordinate exchange, a version of cyclic ascent, is a popular algorithm for finding optimal designs (Meyer and Nachtsheim 1995)

• optimisation of \(\xi = (\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n)\) proceeds coordinate-wise, i.e. just one of the \(x_{ij}\) is varied at a time

Approximate coordinate exchange (ACE) combines coordinate exchange with smoothing to find high-dimensional designs under computationally expensive approximate expected utilities

• a nonparametric regression model (a Gaussian process) is used to smooth the Monte Carlo approximations of \(U(\xi)\) as a function of one coordinate
  • reduces the computational burden
  • facilitates optimisation of a noisy function

Overstall and Woods (2017)

Return to the multifactor logistic regression (crystallography) example

## set up prior
priorMFL <- function(B) {
  b0 <- runif(B, -3, 3)
  b1 <- runif(B, 4, 10)
  b2 <- runif(B, 5, 11)
  b3 <- runif(B, -6, 0)
  b4 <- runif(B, -2.5, 2.5)
  cbind(b0, b1, b2, b3, b4)
}
## define the utility function
MFL.utilSIG <- utilityglm(formula = ~ x1 + x2 + x3 + x4, family = binomial, prior = priorMFL, 
                          criterion = "SIG")$utility
## starting design with n=18 runs, on [-1, 1]
d <- 2 * randomLHS(18, 4) - 1
colnames(d) <- paste0("x", 1:4)

## not run - quite computationally expensive
MLF.ace <- ace(utility = MFL.utilSIG, start.d = d, progress = T)

For this logistic regression example, acebayes has some designs precomputed

pairs(optdeslrsig(18), pch = 16, 
      labels=c(expression(x[1]), expression(x[2]), expression(x[3]), expression(x[4])), cex = 2)

Hierachical logistic regression with \(g=3\) groups (blocks, eg wellplates)

pairs(optdeshlrsig(18), pch = 16, 
      labels=c(expression(x[1]), expression(x[2]), expression(x[3]), expression(x[4])),
      col = c("black", "red", "blue")[rep(1:3, rep(6, 3))], cex = 2)

Reasonably recent overviews of the topics discussed here, and many more, are given in the Handbook of Design and Analysis of Experiments (2015, eds Dean, Morris, Stufken, Bingham; CRC Press).

Some nice examples of recent experiments in technology, economics and social science are described by Luca and Bazerman (2020, The Power of Experiments; MIT Press).

Good online resources include

• Meier’s short intro to ANOVA
• Taback’s design of experiments and observational studies
• Krennrich’s experimental design and process optimisation in R